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be replaced by the resulting set of dimensionless groups, and these can be used to completely define the system behavior. That is, any valid relationship (theoretical or empirical) between the original variables can be expressed in terms of these dimensionless groups. This has two important advantages:

1. Dimensionless quantities are universal (as we have seen), so any relationship involving dimensionless variables is independent of the size or scale of the system. Consequently, information obtained from a model (small-scale) system that is represented in dimensionless form can be applied directly to geometrically and dynamically similar systems of any size or scale. This allows us to translate information directly from laboratory models to large-scale equipment or plant operations (scale-up). Geometrical similarity requires that the two systems have the same shape (geometry), and dynamical similarity requries them to be operating in the same dynamic regime (i.e., both must be either laminar or turbulent). This will be expanded upon later.

2. The number of dimensionless groups is invariably less than the number of original variables involved in the problem. Thus the relations that define the behavior of a given system are much simpler when expressed in terms of the dimensionless variables, because fewer variables are required. In other words, the amount of effort required to represent a relationship between the dimen-sionless groups is much less than that required to relate each of the variables independently, and the resulting relation will thus be simpler in form. For example, a relation between two variables (x vs. y) requires two dimensions, whereas a relation between three variables (x vs. y vs z) requires three dimensions, or a family of two-dimensional "curves" (e.g., a set of x vs. y curves, each curve for a different z). This is equivalent to the difference between one page and a book of many pages. Relating four variables would obviously require many books or volumes. Thus, reducing the number of variables from, say, four to two would dramatically simplify any problem involving these variables.

It is important to realize that the process of dimensional analysis only replaces the set of original (dimensional) variables with an equivalent (smaller) set of dimensionless variables (i.e., the dimensionless groups). It does not tell how these variables are related—the relationship must be determined either theoretically by application of basic scientific principles or empirically by measurements and data analysis. However, dimensional analysis is a very powerful tool in that it can rovide a direct guide for experimental design and scale-up and for expressing operating relationships in the most general and useful form.

There are a number of different approaches to dimensional analysis. The classical method is the "Buckingham n Theorem'', so-called because Buckingham used the symbol n to represent the dimensionless groups. Another classic approach, which involves a more direct application of the law of conservation of dimensions, is attributed to Lord Rayleigh. Numerous variations on these methods have also been presented in the literature. The one thing all of these methods have in common is that they require a knowledge of the variables and parameters that are important in the problem as a starting point. This can be determined through common sense, logic, intuition, experience, or physical reasoning or by asking someone who is more experienced or knowledgeable. They can also be determined from a knowledge of the physical principles that govern the system (e.g., the conservation of mass, energy, momentum, etc., as written for the specific system to be analyzed) along with the fundamental equations that describe these principles. These equations may be macroscopic or microscopic (e.g., coupled sets of partial differential equations, along with their boundary conditions). However, this knowledge often requires as much (or more) insight, intuition, and/or experience as is required to compose the list of variables from logical deduction or intuition. The analysis of any engineering problem requires key assumptions to distinguish those factors that are important in the problem from those that are insignificant. [This can be referred to as the "bathwater'' rule—it is necessary to separate the "baby" from the "bathwater" in any problem, i.e., to retain the significant elements (the "baby") and discard the insignificant ones (the "bathwater''), and not vice versa!] The talent required to do this depends much more upon sound understanding of fundamentals and the exercise of good judgment than upon mathematical facility, and the best engineer is often the one who is able to make the most appropriate assumptions to simplify a problem (i.e., to discard the "bathwater" and retain the "baby"). Many problem statements, as well as solutions, involve assumptions that are implied but not stated. One should always be on the lookout for such implicit assumptions and try to identify them wherever possible, since they set corresponding limits on the applicability of the results.

The method we will use to illustrate the dimensional analysis process is one that involves a minimum of manipulations. It does require an initial knowledge of the variables (and parameters) that are important in the system and the dimensions of these variables. The objective of the process is to determine an appropriate set of dimensionless groups of these variables that can then be used in place of the original individual variables for the purpose of describing the behavior of the system. The process will be explained by means of an example, and the results will be used to illustrate the application of dimensional analysis to experimental design and scale-up.

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