Scale dimensional analysis

Astaiuta, G. Chem. ting. Sci 52 (1997) 4681. Dimensional analysis, scaling and orders of magnitude. Blackman, D. R. SI Units in Engineering (Macmillan, 1969). 

Zlokarnik M. Dimensional Analysis, Scale-Up. In Elickinger MC, Drew SW, eds. Encyclopedia of Bioprocess Technology Eermentation, BioCatalysis, and Bioseparation. New York John Wiley and Sons Inc., 1999 840-861. 

M. Zlokarnik. Dimensional Analysis, Scale-Up. In Encyclopedia of Bio process Technology Fermentation, Biocatalysis, Bioseparation. Vol. 2, 840-861. (M.C. Flickinger and S. W. Drew, eds.) Wiley, 1999. 

Zlokarnik, M. Dimensional analysis, scale-up. In Encyclopedia of Bioprocess Technology Fermentation. Biocatalysis and Bioseparation. Flickinger, M.C., Drew, St. W., eds. Wiley, New York. 1999, pp 840-861. 

Peuikhurst, R.C. Dimensional Analysis Scale Factors. Monographs for Students. Chapman, London (1964)... 

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. 

In addition, dimensional analysis can be used in the design of scale experiments. For example, if a spherical storage tank of diameter dis to be constmcted, the problem is to determine windload at a velocity p. Equations 34 and 36 indicate that, once the drag coefficient Cg is known, the drag can be calculated from Cg immediately. But Cg is uniquely determined by the value of the Reynolds number Ke. Thus, a scale model can be set up to simulate the Reynolds number of the spherical tank. To this end, let a sphere of diameter tC be immersed in a fluid of density p and viscosity ]1 and towed at the speed of p o. Requiting that this model experiment have the same Reynolds number as the spherical storage tank gives... 

Scale- Up of Electrochemical Reactors. The intermediate scale of the pilot plant is frequendy used in the scale-up of an electrochemical reactor or process to full scale. Dimensional analysis (qv) has been used in chemical engineering scale-up to simplify and generalize a multivariant system, and may be appHed to electrochemical systems, but has shown limitations. It is best used in conjunction with mathematical models. Scale-up often involves seeking a few critical parameters. Eor electrochemical cells, these parameters are generally current distribution and cell resistance. The characteristics of electrolytic process scale-up have been described (63—65). 

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 ... 

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. 

When using dimensional analysis in computing or predicting performance based on tests performed on smaller-scale units, it is not physically possible to keep all parameters constant. The variation of the final results will depend on the scale-up factor and the difference in the fluid medium. It is important in any type of dimensionless study to understand the limit of the parameters and that the geometrical scale-up of similar parameters must remain constant. 

Various proposed values for the constants can be found in the literature [8]. Despite double-layer model predictions [148,149] that exponents Jt and y are both unity, and a dimensional analysis model [204] giving x as 1.88 andy as 0.88, test work on a practical scale [202,203] has indicated that both exponents are approximately equal to 2. This implies that a is roughly independent of pipe diameter and that the ratio //3 s 4/jt s 1. 

Dickey, D.S., 1993. Dimensional analysis, similarity and scale-up. American Institute of Chemical Engineers Symposium Series, 293, 143-150. 

Zlokaniik, M., 1991. Dimensional Analysis and Scale-up in Chemical Engineering. Spriiiger-Verlag. 

These scale-up methods will necessarily at times include fundamental concepts, dimensional analysis, empirical correlations, test data, and experience [32]. 

Dimensional analysis techniques are especially useful for manufacturers that make families of products that vary in size and performance specifications. Often it is not economic to make full-scale prototypes of a final product (e.g., dams, bridges, communication antennas, etc.). Thus, the solution to many of these design problems is to create small scale physical models that can be tested in similar operational environments. The dimensional analysis terms combined with results of physical modeling form the basis for interpreting data and development of full-scale prototype devices or systems. Use of dimensional analysis in fluid mechanics is given in the following example. 

When applied to the relaxation time of a polymer, dimensional analysis of Eq. (22) shows that the following scaling transformation should be written for tr ... 

Chemical engineering analysis requires the formulation of relationships which will apply over a wide range of size of the individual items of a plant. This problem of scale up is vital and it is much helped by dimensional analysis. 

On the basis of scaling arguments, general functional dependencies can also be derived. For example, dimensional analysis shows that the center of mass diffusion coefficient DG for Zimm relaxation has the form... 

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. 

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