Dimensional Analysis and Scaling Rules

In this chapter we shall derive and present relationships or formulae that will allow us to predict a cyclone s cut-point diameter, grade-efficiency curve, overall or gross efficiency, and pressure drop on the basis of measurements taken on a geometrically similar cyclone. These formulae should also allow us to evaluate the performance of an operating cyclone and, if necessary, assist us in troubleshooting its design, mechanical condition, or mode of operation. 

When scaling cyclones we have to consider not only the fluid but also the particle dynamics. This might lead us to expect complicated scaling laws, but in the end we shall find that simple rules can provide a wealth of useful information. 

To derive the dimensionless groups for cyclones, we can proceed along two lines of inquiry  

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In the development of these processes and their transference into an industrial-scale, dimensional analysis and scale-up based on it play only a subordinate role. This is reasonable, because one is often forced to perform experiments in a demonstration plant which copes in its scope with a small produdion plant ( mock-up plant, ca. 1/10-th of the industrial scale). Experiments in such plants are costly and often time-consuming, but they are often indispensable for the lay-out of a technical plant. This is because the experiments performed in them deliver a valuable information about the scale-dependent hydrodynamic behavior (arculation of liquids and of dispersed solids, residence time distributions). As model substances hydrocarbons as the liquid phase and nitrogen or air as the gas phase are used. The operation conditions are ambient temperature and atmospheric pressure ( cold-flow model ). As a rule, the experiments are evaluated according to dimensional analysis. 

In this section, a summary will be provided about how dimensional analysis can be properly utilized in order to obtain, with a minimum of time and expenditure, the results necessary for the determination of both the pi-relationship (process characteristic) of the process in question and a valid scale-up rule. In this context, nothing new will be imparted. All of this information has already been disclosed in previous chapters. 

In general, the university graduate has not at all been adequately trained to deal with such problems. On the one hand, treatises on dimensional analysis, the theory of similarity and scale-up methods included in common, run of the mill textbooks on chemical engineering are out of date. In addition, they are only seldomly written in such a manner that would popularize these methods. On the other hand, there is no motivation for this type of research at universities since, as a rule, they are not confronted with scale-up tasks and are therefore not equipped with the necessary apparatus on the bench-scale. 

The rules given by Damkohler (Dl) for changing the scale of catalytic reactors without changing the course of the reaction were derived primarily by dimensional analysis. A better idea of the requirements for scaling up can be obtained by a detailed examination of the coefficients in the differential equations and boundary conditions describing the reactor, with the independent variables in the equations transformed to a modified reciprocal space velocity and a dimensionless radial variable. In an exact scale model, these coefficients are all the same as they are in... 

As a rule process engineering measurements in the laboratory are carried out with the aim of supplying design data for the industrial plant. For this purpose it is required that a complete similarity exists between the process on laboratory-scale and that on industrial-scale, which is expressed in identical numerical values for the process-describing numbers Hi = idem (see Section 1.6, Dimensional analysis). 

Multi-mode pushover analysis procedure IRSA (Incremental Response Spectrum Analysis) has been introduced by the first author to enable the two and three dimensional nonlinear analyses of buildings and bridges (Aydmoglu 2003). The practical version of the procedure (Aydmoglu 2004, 2007) works directly with smoothed elastic response spectrum and makes use of the well-known equal displacement rule to scale modal displacement increments at each piecewise linear step of an incremental application of linear Response Spectrum Analysis (RSA). In this paper, main steps of IRSA are summarized and its performance is evaluated on two example bridges under three different ground motions. 

The rules that govern polymer epitaxy have been deduced mainly from the analysis of observed epitaxies. The dimensional match at the unit-cell scale merely implies that the local favorable interaction are repeated on a two-dimensional (or at least on a one-dimensional) array in the contact plane. The topographic matching refers to the unit-cell scale interactions. In some instances (illustrated earlier with, e.g., the formation of the sPP Form II on 2-quinoxaUnol) the topographic requirements are obvious and indeed intuitively sound. 

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