Scaling and Dimensionless Groups

Dimensionless variables consisting of the dependent variables and the time and space coordinates will be designated by (A). Properties and other constants that form 

Fundamentals of Fire Phenomena James G. Quintiere 2006 John Wiley Sons, Ltd ISBN 0 470 09113-4 

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We now discuss the basic features of the RG flow. This amounts to giving a physical interpretation of our general discussion of the dilatation group, and it explains qualitatively the characteristic features like universality, power laws, and scaling observed in a critical system. For a given class of systems (fluids, uniaxial ferromagnets, polymer solutions, etc.) we envisage the space of all Hamiltonians to be parameterized by a microscopic scale and dimensionless... 

K) Principle of Similarity and Dimensionless Groups. This principle states that experimental data obtained in a lab-sized vessel can be used to predict the equipment requirements to get satisfactory performance in similar industrial-scale equipment. There are three types of similitude ... 

Three basic approaches have been used to solve the equations of motion. For relatively simple configurations, direct solution is possible. For complex configurations, numerical methods can be employed. For many practical situations, particularly three-dimensional or one-of-a-kind configurations, scale modeling is employed and the results are interpreted in terms of dimensionless groups. This section outlines the procedures employed and the limitations of these approaches (see Computer-aided engineering (CAE)). 

Dimensional Analysis. Dimensional analysis can be helpful in analyzing reactor performance and developing scale-up criteria. Seven dimensionless groups used in generalized rate equations for continuous flow reaction systems are Hsted in Table 4. Other dimensionless groups apply in specific situations (58—61). Compromising assumptions are often necessary, and their vaHdation must be estabHshed experimentally or by analogy to previously studied systems. 

For purposes of data correlation, model studies, and scale-up, it is useful to arrange variables into dimensionless groups. Table 6-7 lists many of the dimensionless groups commonly founa in fluid mechanics problems, along with their physical interpretations and areas of application. More extensive tabulations may oe found in Catchpole and Fulford (Ind. Eng. Chem., 58[3], 46-60 [1966]) and Fulford and Catchpole (Ind. Eng. Chem., 60[3], 71-78 [1968]). 

Turbomachines can be compared with each other by dimensional analysis. This analysis produces various types of geometrically similar parameters. Dimensional analysis is a procedure where variables representing a physical situation are reduced into groups, which are dimensionless. These dimensionless groups can then be used to compare performance of various types of machines with each other. Dimensional analysis as used in turbomachines can be employed to (1) compare data from various types of machines—it is a useful technique in the development of blade passages and blade profiles, (2) select various types of units based on maximum efficiency and pressure head required, and (3) predict a prototype s performance from tests conducted on a smaller scale model or at lower speeds. 

The objectives are not realized when physical modeling are applied to complex processes. However, consideration of the appropriate differential equations at steady state for the conservation of mass, momentum, and thermal energy has resulted in various dimensionless groups. These groups must be equal for both the model and the prototype for complete similarity to exist on scale-up. 

Of the possible sets of scaling laws containing only the dimensionless group of properties pjpv, set 4, which includes A, pL, pv, y, C L, and kL, appears closest to consistency with the available experimental evidence for burn-out in uniformly heated round tubes. 

Thus, the ratios of the various forces occurring in mixing vessels can be expressed as the above dimensionless groups which, in turn, serve as similarity parameters for scale-up of mixing equipment. It can be shown that the existence of geometric and dynamic similarities also ensures kinematic similarity. 

Equations (3.11) and (3.12) show that the friction factor of a rectangular micro-channel is determined by two dimensionless groups (1) the Reynolds number that is defined by channel depth, and (2) the channel aspect ratio. It is essential that the introduction of a hydraulic diameter as the characteristic length scale does not allow for the reduction of the number of dimensionless groups to one. We obtain... 

An alternative to the measurement of the dimensions of the indentation by means of a microscope is the direct reading method, of which the Rockwell method is an example. The Rockwell hardness is based on indentation into the sample under the action of two consecutively applied loads - a minor load (initial) and a standardised major load (final). In order to eliminate zero error and possible surface effects due to roughness or scale, the initial or minor load is first applied and produce an initial indentation. The Rockwell hardness is based on the increment in the indentation depth produced by the major load over that produced by the minor load. Rockwell hardness scales are divided into a number of groups, each one of these corresponding to a specified penetrator and a specified value of the major load. The different combinations are designated by different subscripts used to express the Rockwell hardness number. Thus, when the test is performed with 150 kg load and a diamond cone indentor, the resulting hardness number is called the Rockwell C (Rc) hardness. If the applied load is 100 kg and the indentor used is a 1.58 mm diameter hardened steel ball, a Rockwell B (RB) hardness number is obtained. The facts that the dial has several scales and that different indentation tools can be filled, enable Rockwell machine to be used equally well for hard and soft materials and for small and thin specimens. Rockwell hardness number is dimensionless. The test is easy to carry out and rapidly accomplished. As a result it is used widely in industrial applications, particularly in quality situations. 

A current example of a problem that can be simplified through segregation of its components by physical scale is the deposition of on-chip interconnects onto a wafer. Takahashi and Gross have analyzed the scaling properties of interconnect fabrication problems and identified the relevant control parameters for the different levels of pattern scale [135], They define several dimensionless groups which determine the type of problem that must be solved at each level. 

On the submicron scale, the current distribution is determined by the diffusive transport of metal ion and additives under the influence of local conditions at the interface. Transport of additives in solution may be non-locally controlled if they are consumed at a mass-transfer limited rate at the deposit surface. The diffusion of additives in solution must then be solved simultaneously with the flux of reactive ion. Diffusive transport of inhibitors forms the basis for leveling [144-147] where a diffusion-limited inhibitor reduces the current density on protrusions. West has treated the theory of filling based on leveling alone [148], In his model, the controlling dimensionless groups are equivalent to and D divided by the trench aspect ratio. They determine the ranges of concentration within which filling can be achieved. 

The principle of similarity [Holland (1964), Johnstone and Thring (1957)] together with the use of dimensionless groups is the essential basis of scale-up. The types of similarity relevant to liquid mixing systems together with their definitions are listed as follows. 

Each dimensionless group represents a rule for scale-up. Frequently these individual scale-up rules conflict. For example, scale-up on dynamic similarity should depend chiefly upon a single dimensionless group that represents the ratio of the applied to the opposing forces. The Reynolds, Froude and Weber numbers are the ratios of the applied to the resisting viscous, gravitational and surfaces forces, respectively. 

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