Classical Dimensional Analysis

In a) we list the variables influencing the cyclone performance, and arrange them in dimensionless groups. In b) we arrive at the groups by making the equations of motion for the gas and the particles dimensionless. Both lines of enquiry are enlightening in their own way, so we shall follow both, the latter in Appendix 8.A. 

The separation efficiency in a cyclone depends on a series of physical and 

This is a large munber of parameters, the full list is  

In order to render the process more tractable, we must make some simplifications and assmnptions  

These simplifications and assumptions do not significantly limit the range of application of our analysis. We now make some additional assumptions, however, that do limit the range of systems to which our analysis can be applied. The reader confronted with a specific scaling problem should check which, if any, of the assumptions below are warranted for his or her system. 

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This is as far as classical dimensional analysis can take us. However, in Appendix 8. A we obtain more physical insight by inspecting the equations of motion for the gas and the particles. One important result of this is that the density ratio in (8.1.5) need not appear separately, as the effect of the particle density is accounted for in Stk. This fact allows us to simplify (8.1.5) even further so that it becomes ... 

This is the formal requirement for dynamic similarity, and is consistent with the results of the classical dimensional analysis in the main text. As we mentioned there, experience teaches us that over a wide range of operating conditions Reynolds number similarity is not all that critical for Stokes number similarity between cyclones, and this indicates that, in this range, it is not all that critical for dynamic similarity. 

Inspection of the equations of motion of the gas and particle phases has thus confirmed the results of classical dimensional analysis, simplified the results of the analysis further, and has, we trust, increased our understanding of the physical significance of the dimensionless groups. 

Along with the methods of similarity theory, Ya.B. extensively used and enriched the important concept of self-similarity. Ya.B. discovered the property of self-similarity in many problems which he studied, beginning with his hydrodynamic papers in 1937 and his first papers on nitrogen oxidation (25, 26). Let us mention his joint work with A. S. Kompaneets [7] on selfsimilar solutions of nonlinear thermal conduction problems. A remarkable property of strong thermal waves before whose front the thermal conduction is zero was discovered here for the first time their finite propagation velocity. Independently, but somewhat later, similar results were obtained by G. I. Barenblatt in another physical problem, the filtration of gas and underground water. But these were classical self-similarities the exponents in the self-similar variables were obtained in these problems from dimensional analysis and the conservation laws. 

The first problem is the classical example used to show the scientific force of the dimensional analysis - and especially of the pi theorem. Remember that we are interested in the pressure drop per unit length (Ap/1) along the pipe. According to the experimenter s knowledge of the problem and to step 1, we must list all the pertinent variables that are involved in this problem, it was assumed that ... 

The classic example of a distorted model occurs in the study of liquid media which are mixed mechanically as described earlier in this chapter. The dimensional analysis shows that ... 

The dependence of Vik on mass may now be obtained from a dimensional analysis. The friction coefficient rik depends on two parameters defining the state of the system, say kT and the overall particle concentration c, and on the parameters of the cited equation of relative motion. These parameters are the reduced mass niik = rriimk/ rrii -f rrik) and a set of ( -values corresponding to a set of d,fc-values. Planck s constant h does not enter into our classical calculation. The dimensions of the quantities involved are given in Table IV. 

Classical techniques have relied heavily on dimensional analysis, the combining of the many variables into physically meaningful non-dimensional groups, supported with experiments to quantify heat transfer for various geometries. For most drying applications of pharmaceutical relevance, the most important of these non-dimensional groups are the Nusselt number (Nu), the Prandtl number (Pr) and the Reynolds number (Re), defined as follows ... 

The dimensionless equation describing the transfer phenomena may be obtained either by direct reference to the ratios of the physical quantities or by recourse to the classical techniques of dimensional analysis, i.e., the Buckingham n Theorem or Rayleigh s method of indices. In addition, the basic differential equations governing the process may be reduced to dimensionless form and the coefficients identified. In general, the dimensionless equation for heat transfer through the combined film is... 

There are two classical methods in dimensional analysis, Buckingham s pi theorem and the method of indices by Lord Rayleigh. Here we will briefly explain the more common of the two Buckingham s theorem. 

In spite of the wide variety of interfacial media encountered, it is possible in many cases to obtain surface balance equations, which are representative and unrestrictive in comparison to actual phenomena. The constitutive laws can be established by dimensional analysis. They relate to surface variables, to flux and rates of production. The phenomenological coefficients can be determined experimentally, or by detailed analysis over the interfacial thickness. The solution implies a coupling between these equations and those for volumes in contact. In the examples presented here, the results are in perfect agreement with those of simple classical theory. 

Engineering Problem Solving A Classical Perspective 8.3 Dimensional Analysis... 

In the years following 1923, research papers dealt with the measurement of viscosity as a function of concentration, temperature, and pressure, along with a variety of experimental studies based on dimensional analysis. A classic example of special interest is the three part series by A.P. Colburn and co-workers. The titles of the three papers are ... 

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