Scaling rule

The first scale-up method was developed by Carley and McKelvey [1] in 1953. In this original work, the rate and power consumption at a fixed speed and for a geometrically similar extruder was simply a ratio of the extruder diameters as follows  

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Figure 5. Schematic diagram of manufactured tube, showing the scaling rule and in (a) the area that was captured by the shearography process in the following figures.

Detailed economics of individual industrial processes, including SCP processes, are usually regarded as confidential, out of fear that publication may lend advantage to competitors. In addition, economy of scale rule generally applies (that is as the production capacity increases, the cost of the product decreases), so that direct comparisons can only be made between systems of similar capacity. Some economic data on SCP processes have been published and are presented in the Resource Material at the end of this chapter. You should appreciate that the data are outdated by more than a decade, during which time substrate costs will have varied relative to each other, and technology will have improved. This means that the comparative costs presented in Table 4.13, for example, may not be now as presented there. Nevertheless the data presented do provide an outline of the economics of SCP production. The processes referred to in the Resource Material are not necessarily those mentioned in the text and so you may find some differences in detail. 

Equipment costs are sometimes estimated using a scaling rule ... 

Hashimoto T., Shibayama M., and Kawai H., Ordered structure in block copolymer solution. 4. Scaling rules on size of fluctuations with block molecular weight, concentration temperature in segregation and homogeneous regimes. Macromolecules, 16, 1093, 1983. 

Beever, P. F., "Scaling Rules for Prediction of Thermal Ruanway," in Proceedings of the International Symposium on Runaway Reactions, p. 1, Center for Chemical Process Safety/AIChE, New York, NY (1989). 

According to the scaling rules, the MOSFET transconductance is expected to remain constant if both the lateral and vertical device dimensions are reduced by the same scaling factor A. For a short-channel MOSFET, the saturation current Idsat may indeed be expressed as... 

This analysis therefore provides some theoretical justification for the rather empirical scaling rules which have been used to simulate the effect of orthogonalisation in earlier work (5,24). 

Scaling rules and ratios have also been developed to guide the screw designer. These rules have many different foundations, but they all attempt to maintain a design parameter constant with the scaling. For example, it might be desirable to maintain a constant discharge temperature and scale from a 100 mm diameter... 

Cash L, Cayias JL, Pournier G, MacAllister D, Shares T, Schechter RS, Wade WH (1977) The application of low interfacial tension scaling rules to binary hydrocarbon mixtures. J Colloid Interface Sci 59 39-44... 

Until rigorous scale-up rules are determined, these cautionary rules are the state of the art for now. We offer a first step toward rigorous scaling rules by scaling particle surface velocities but caution that this work is only preliminary in nature. The best advice is to be cautious—understand the physics behind the problem and that statistics of the data collected. Remember that a fundamental understanding of the issues is still limited and luck is unlikely to be on your side, hence frustrating trial-and-error is still likely (and unfortunately) necessary to be employed. 

When two signals are added together, the worst case word growth in two s complement number representation would be one bit. To prevent the possibility of overflow, N channels added together would be scaled by l IN, which corresponds to using an LI Norm scaling rule [Oppenheim and Schafer, 1989]. 

Although the above solution satisfies our requirements, it leads to a very small volumetric throughput. However, in industry there are various scaling rules that are used for extruder systems which compromise one or the other requirement. We cover this in more detail in Chapter 4 of this book. 

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