Dimensional analysis theory

H. L. Langhaar, Dimensional Analysis and Theory ofModelSs oEn Wiley Sons, Inc., New York, 1951. 

Theoretically based correlations (or semitheoretical extensions of them), rooted in thermodynamics or other fundamentals are ordinarily preferred. However, rigorous theoretical understanding of real systems is far from complete, and purely empirical correlations typically have strict limits on apphcabihty. Many correlations result from curve-fitting the desired parameter to an appropriate independent variable. Some fitting exercises are rooted in theory, eg, Antoine s equation for vapor pressure others can be described as being semitheoretical. These distinctions usually do not refer to adherence to the observations of natural systems, but rather to the agreement in form to mathematical models of idealized systems. The advent of readily available computers has revolutionized the development and use of correlation techniques (see Chemometrics Computer technology Dimensional analysis). 

Bridgman had strong views on the importance of empirical research, influenced as little as possible by theory, and this helped him test the influence of numerous variables that lesser mortals failed to heed. He kept clear of quantum mechanics and dislocation theory, for instance. He became deeply ensconced in the philosophy of physics research for instance, he published a famous book on dimensional analysis, and another on the logic of modern physics . When he sought to extrapolate his ideas into the domain of social science, he found himself embroiled in harsh disputes this has happened to a number of eminent scientists, for instance, J.D. Bernal. Walter s book goes into this aspect of Bridgman s life in detail. 

Langhaar, H. L., Dimensional Analysis and Theory of Models, John Wiley and Sons, 1951. 

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... 

A mathematical analysis of the process is extremely difficult and requires to solve the Reynolds equation of lubrication theory and apply the solution to the cavitation boundary conditions. A two-dimensional analysis of the pressure distribution in the plane of the roll nip showed that the liquid pressure rises sharply to a large value near the nip, and drops equally sharply to a minimum justbeyond the nip. Before large negative pressures are reached, the liquid may cavitate as a result of the expansion of entrained gases within the liquid. 

We have shown that the vector mesons in the CFL phase have masses of the order of the color superconductive gap, A. On the other hand the solitons have masses proportional to F%/A and hence should play no role for the physics of the CFL phase at large chemical potential. We have noted that the product of the soliton mass and the vector meson mass is independent of the gap. This behavior reflects a form of electromagnetic duality in the sense of Montonen and Olive [29], We have predicted that the nucleon mass times the vector meson mass scales as the square of the pion decay constant at any nonzero chemical potential. In the presence of two or more scales provided by the underlying theory the spectrum of massive states shows very different behaviors which cannot be obtained by assuming a naive dimensional analysis. 

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

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. 

Dimensional analysis is a method for producing dimensionless numbers that completely characterize the process. The analysis can be applied even when the equations governing the process are not known. According to the theory of models, two processes may be considered completely similar if they take place in similar geometrical space and if all the dimensionless numbers necessary to describe the process have the same numerical value (2). 

M. Zlokarnik. Dimensional analysis and scale-up in theory and industrial apphcation. In M. Levin, ed. Pharmaceutical Process Scale-Up. New York Marcel Dekker, 2001. 

It appears that the formal theories are not sufficiently sensitive to structure to be of much help in dealing with linear viscoelastic response Williams analysis is the most complete theory available, and yet even here a dimensional analysis is required to find a form for the pair correlation function. Moreover, molecular weight dependence in the resulting viscosity expression [Eq. (6.11)] is much too weak to represent behavior even at moderate concentrations. Williams suggests that the combination of variables in Eq. (6.11) may furnish theoretical support correlations of the form tj0 = f c rjj) at moderate concentrations (cf. Section 5). However the weakness of the predicted dependence compared to experiment and the somewhat arbitrary nature of the dimensional analysis makes the suggestion rather questionable. 

Along with the methods of similarity theory, Ya.B. extensively used and enriched the important concept of self-similarity. Ya.B. discovered the property of self-similarity in many problems which he studied, beginning with his hydrodynamic papers in 1937 and his first papers on nitrogen oxidation (25, 26). Let us mention his joint work with A. S. Kompaneets [7] on selfsimilar solutions of nonlinear thermal conduction problems. A remarkable property of strong thermal waves before whose front the thermal conduction is zero was discovered here for the first time their finite propagation velocity. Independently, but somewhat later, similar results were obtained by G. I. Barenblatt in another physical problem, the filtration of gas and underground water. But these were classical self-similarities the exponents in the self-similar variables were obtained in these problems from dimensional analysis and the conservation laws. 

However, for our purposes this method is too complicated, and we will obtain the desired results (albeit accurate only to within unknown numerical factors) using the methods of dimensional analysis and the theory of similarity. 

The bust line of Eq. (7.9) follows since dimensionless quantities can depend only on dimensionless combinations of their variables. As a result of this simple dimensional analysis in the continuous chain model we have found the two parameter theory instead of the three independent parameters ,3A.nt physical observables involve only the two combinations Rq, z. Perturbation theory now proceeds in powers of 2. Thus the continuous chain limit gives a precise meaning to the simple argument presented in Chap. 6. 

This transformation leaves both jRq and 2 = Pen 2 (Eq. (7.10)) invariant. It just expresses naive dimensional analysis in the continuous chain model. The power of the RG-approach lies in the fact that we can construct nontrivial realizations. These take into account more than just the leading n-dependence of each order of perturbation theory and therefore obey the condition of invariance of the macroscopic observables up to much smaller corrections. 

The latter assumption distinguishes scaling theory from straightforward dimensional analysis in the continous chain model. 

Chapter 7 covers the kinetic theory of gases. Diffusion and the one-dimensional velocity distribution were moved to Chapter 4 the ideal gas law is used throughout the book. This chapter covers more complex material. I have placed this material later in this edition, because any reasonable derivation of PV = nRT or the three-dimensional speed distribution really requires the students to understand a good deal of freshman physics. There is also significant coverage of dimensional analysis determining the correct functional form for the diffusion constant, for example. 

It is usually impossible to determine all the essential facts for a given fluid flow by pure theory, and hence much dependence must be placed on experimental investigations. The number of tests to be made can be greatly reduced by a systematic program based on dimensional analysis and specifically on the laws of similitude or similarity, which means certain relations by which test data can be applied to other cases. 

A. Carpinteri et al Cohesive crack model description of ductile to brittle size-scale transition dimensional analysis vs. renormalization group theory. Eng. Fract. Mech. 70(14) 1809-1839 (2003)... 

---