Applications of Dimensional Analysis

Here we illustrate the use of dimensional analysis to determine the relationship between the drag force, F, on a smooth sphere and the affecting physical quantities, namely, the diameter of the sphere, D, the fluid velocity, V, density, p, and viscosity, p. It is assumed that f D,V,F,p, i) = 0. The dimensional matrix is formed as follows  

To make the groups of all the physical quantities dimensionless, it is convenient to form the products as From the matrix, the following equations are formed  

Experimental data are, however, correlated as = (8/7r)7Ti, known as the drag coefficient, while Ti2 is the well-known Reynolds number. 

When the experimental data were plotted using log(xP/m) as a function of a 

Software Dimensional Analysis Toolbox in MatLab, written by Steffen Bruckner, may be available free of cost for academic use. It may be obtained from http //www.sbrs.net/along with the usage instruction. The user is referred to the license policy prior to use. 

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Another empirical expression that can be used to predict pressure drop, again based upon application of dimensional analysis is given by the following formula. This relationship is applicable to fabrics having porosities over the range of 0.88 to 0.96,... 

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 ... 

Application of dimensional analysis to Equation (8) using a set of fundamental units containing heat, mass, length, time, and temperature, yields... 

Scaling. The fact that the value of the dimensionless parameter a is the same regardless of the units (e.g., scale) used in the problem illustrates the universal nature of dimensionless quantities. That is, the magnitude of any dimensionless quantity will always be independent of the scale of the problem or the system of (consistent) units used. This is the basis for the application of dimensional analysis, which permits information and relationships determined in a small-scale system (e.g., a model ) to be applied directly to a similar system of a different size if the system variables are expressed in dimensionless form. This process is known as scale-up. 

As an example of the application of dimensional analysis to experimental design and scale-up, consider the following example. 

The application of dimensional analysis is indeed heavily dependent on the available knowledge. The following five steps (Fig. 5) can be outlined as ... 

All the relevant physical variables describing the problem are known. —> The application of dimensional analysis is unproblematic. 

It must, of course, be said that approaching a problem from the point of view of dimensional analysis also remains useful even if all the variables relevant to the problem are not yet known The timely application of dimensional analysis may often lead to the discovery of forgotten variables or the exclusion of artifacts. 

Additional insights into the application of dimensional analysis to scale-up can be found in the chapter in this volume by Zlokarnik (65) and in his earlier monograph on scale-up in chemical engineering (66). 

Zlokarnik, M. Problems in the application of dimensional analysis and scale-up of mixing operations. Chem Eng Sci 1998 53(17) 3023-3030. 

Application of dimensional analysis to Eq. (42) leads to the dimensionless groups of the Dittus-Boelter equation for Eq. (43), one obtains... 

Dimensional analysis is a useful tool for examining complex engineering problems by grouping process variables into sets that can be analyzed separately. If appropriate parameters are identified, the number of experiments needed for process design can be reduced, and the results can be described in simple mathematical expressions. In addition, the application of dimensional analysis may facilitate the scale-up for selected biotechnology unit operations. A detailed description of dimensional analysis is reviewed by Zlokarnik [18]. 

Giffen (6C) and Mock and Ganger (ISC) present the effect of various operating parameters on fuel atomization in sprays. Applications of dimensional analysis to spray-nozzle performance data are discussed by Shafer and Bovey (23C). 

In fact, the applicability of dimensional analysis depends heavily on the knowledge available. The following five steps can be outlined, see also Fig. 6. 

Fig. 6 Applicability of dimensional analysis, as dependent on the knowledge available afterJ. Pawlowski [26]...

The unavailability of the model material systems can sometimes limit the application of dimensional analysis. In such cases it is of course absolutely wrong to speak of limits of the dimensional analysis . 

The continuity equation (8.9) and the energy equation (8.12) are identical to those for forced convective flow. The x- and y-momentum equations, i.e., Eqs. (8.10) and (8.11), differ, however, from those for forced convective flow due to the presence of the buoyancy terms. The way in which these terms are derived was discussed in Chapter 1 when considering the application of dimensional analysis to convective heat transfer. In these buoyancy terms, is the angle that the x-axis makes to the vertical as shown in Fig. 8.3. 

In fact the application of dimensional analysis is strongly dependent upon the knowledge available. Pawlowski [in 638] formulated the following five levels ... 

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