Dimensional analysis applications

For a comprehensive inventory of dimensional analysis applications in chemical engineering see Dobre and Sanchez Marcano [2007]. For applications in other disciplines see Palacios [1964]... 

Feiste, K. Hanke, R. Stegemann, D. R.eimche, W. Three Dimensional Analysis of Growing Casting Defects. International Symposium on Computerized Tomography for Industrial Applications, Applications II 20, Berlin, 1994. 

Damkdhler (1936) studied the above subjects with the help of dimensional analysis. He concluded from the differential equations, describing chemical reactions in a flow system, that four dimensionless numbers can be derived as criteria for similarity. These four and the Reynolds number are needed to characterize reacting flow systems. He realized that scale-up on this basis can only be achieved by giving up complete similarity. The recognition that these basic dimensionless numbers have general and wider applicability came only in the 1960s. The Damkdhler numbers will be used for the basis of discussion of the subject presented here as follows ... 

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... 

The application of the principles of dimensional analysis may best be understood by considering an example. 

Clearly, the maximum degree of simplification of the problem is achieved by using the greatest possible number of fundamentals since each yields a simultaneous equation of its own. In certain problems, force may be used as a fundamental in addition to mass, length, and time, provided that at no stage in the problem is force defined in terms of mass and acceleration. In heat transfer problems, temperature is usually an additional fundamental, and heat can also be used as a fundamental provided it is not defined in terms of mass and temperature and provided that the equivalence of mechanical and thermal energy is not utilised. Considerable experience is needed in the proper use of dimensional analysis, and its application in a number of areas of fluid flow and heat transfer is seen in the relevant chapters of this Volume. 

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. 

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... 

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

The scope of coverage includes internal flows of Newtonian and non-Newtonian incompressible fluids, adiabatic and isothermal compressible flows (up to sonic or choking conditions), two-phase (gas-liquid, solid-liquid, and gas-solid) flows, external flows (e.g., drag), and flow in porous media. Applications include dimensional analysis and scale-up, piping systems with fittings for Newtonian and non-Newtonian fluids (for unknown driving force, unknown flow rate, unknown diameter, or most economical diameter), compressible pipe flows up to choked flow, flow measurement and control, pumps, compressors, fluid-particle separation methods (e.g.,... 

The Britter and McQuaid10 model was developed by performing a dimensional analysis and correlating existing data on dense cloud dispersion. The model is best suited for instantaneous or continuous ground-level releases of dense gases. The release is assumed to occur at ambient temperature and without aerosol or liquid droplet formation. Atmospheric stability was found to have little effect on the results and is not a part of the model. Most of the data came from dispersion tests in remote rural areas on mostly flat terrain. Thus the results are not applicable to areas where terrain effects are significant. 

The Britter-McQuaid model is a dimensional analysis technique, based on a correlation developed from experimental data. However, the model is based only on data from flat rural terrain and is applicable only to these types of releases. The model is also unable to account for the effects of parameters such as release height, ground roughness, and wind speed profiles. 

The ability of a hammer to deform metal depends on the energy it is able to deliver on impact. Consider a steam hammer (see Fignre 7.9) that has a falling weight of 200 lb and a steam bore, d, equal to 12 in. Assume that the mean effective steam pressure, P, is 80 psi and that the stroke is 30 in. If the hammer travels 1/8 in. into the metal after striking it, determine the average force exerted on the workpiece. You should be able to do this without an equation, through the application of some dimensional analysis. 

Dimensional Analysis and Scale-Up in Theory and Industrial Application... 

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). 

Houcine I, Plasari E, David R, Villermaux J. Feedstream jet intermittency phenomenon in a continuous stirred tank reactor. Chem Eng J 1999 72 19-29. Zlokarnik M. Dimensional analysis and scale-up in theory and industrial application. In Levin M, ed. Process Scale-Up in the Pharmaceutical Industry. New York Marcel Dekker, 2001. 

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

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