Dimensional analysis relationships between dimensionless

We have stated that dimensional analysis results in an appropriate set of groups that can be used to describe the behavior of a system, but it does not tell how these groups are related. In fact, dimensional analysis does not result in any numbers related to the groups (except for exponents on the variables). The relationship between the groups that represents the system behavior must be determined by either theoretical analysis or experimentation. Even when theoretical results are possible, however, it is often necessary to obtain data to evaluate or confirm the adequacy of the theory. Because relationships between dimensionless variables are independent of... 

Since dimensional analysis can provide only a functional (implicit) relationship between dimensionless numbers, Eqs. (5.132) and (5.133) are synonymous dimensionless results. That is, by suitable transformations, a dimensionless result can be made identical to... 

The result is a modified Euler number. You can prove to yourself that the pressure drop over the particle can be obtained by accounting for the projected area of the particle through particle size, S, in the denominator. Thus, by application of dimensional analysis to the force balance expression, a relationship between the dimensionless complexes of the Euler and Reynolds numbers, we obtain ... 

It is important to recognise the differences between scalar quantities which have a magnitude but no direction, and vector quantities which have both magnitude and direction. Most length terms are vectors in the Cartesian system and may have components in the X, Y and Z directions which may be expressed as Lx, Ly and Lz. There must be dimensional consistency in all equations and relationships between physical quantities, and there is therefore the possibility of using all three length dimensions as fundamentals in dimensional analysis. This means that the number of dimensionless groups which are formed will be less. 

The procedure for performing a dimensional analysis will be illustrated by means of an example concerning the flow of a liquid through a circular pipe. In this example we will determine an appropriate set of dimensionless groups that can be used to represent the relationship between the flow rate of an incompressible fluid in a pipeline, the properties of the fluid, the dimensions of the pipeline, and the driving force for moving the fluid, as illustrated in Fig. 2-1. The procedure is as follows. 

It should be emphasized that the specific relationship between the variables or groups that is implied in the foregoing discussion is not determined by dimensional analysis. It must be determined from theoretical or experimental analysis. Dimensional analysis gives only an appropriate set of dimensionless groups that can be used as generalized variables in these relationships. However, because of the universal generality of the dimensionless groups, any functional relationship between them that is valid in any system must also be valid in any other similar system. 

Dimensional analysis is an analytical method wherein a number of experimental variables that govern a given physical phenomenon reduce to form a smaller number of dimensionless variables. A phenomenon which varies as a function of k independent parameters can be reduced to a relationship between (k — m) dimensionless parameters, where m is the number of dimensionally independent parameters. 

Buckinghams 7r-theorem [i] predicts the number of -> dimensionless parameters that are required to characterize a given physical system. A relationship between m different physical parameters (e.g., flux, - diffusion coefficient, time, concentration) can be expressed in terms of m-n dimensionless parameters (which Buckingham dubbed n groups ), where n is the total number of fundamental units (such as m, s, mol) required to express the variables. For an electrochemical system with semiinfinite linear geometry involving a diffusion coefficient (D, units cm2 s 1), flux at x = 0 (fx=o> units moles cm-2 s 1), bulk concentration (coo> units moles cm-3) and time (f, units s), m = 4 (D, fx=0, c, t) and n - 3 (cm, s, moles). Thus m-n - 1 therefore only one dimensionless parameter can be constructed and that is fx=o (t/Dy /coo. Dimensional analysis is a powerful tool for characterizing the behavior of complex physical systems and in many cases can define relationships... 

The surface area to volume ratio of a sphere is used in several models in this book. Dimensional analysis can be used to find the relationship between the surface area (A) with dimensions of [L ] and the internal volume (V) with dimensions of [D for any convex regular solid. The surface area is related to the volume by a dimensionless constant, b. 

The dimensional analysis gives basic relationships between the above-mentioned dimensionless groups, which are useful to describe hydrocyclone operation and are the basis for scale-up calculations aimed at adapting results from laboratory experimentation to an industrial scale for a number of processes in diverse industries. For example, for feed suspensions up to 10% by volume Medronho and Svarovsky (1984) proposed the following relations, for hydrocyclones following Rietema s optimum proportions and treating inert solids suspensions ... 

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