A General Method for Scaling and Analyzing Transport Processes

Abstract

This paper presents a general method for scaling and analyzing transfer processes in physical sciences, natural sciences (biology and ecology,) economics, etc. It is based upon and expresses the principle of equal fractional change.

The formulation rests on three metrics: one for the process transfer rate; the second for the quantity affected by the process; and the third for the time of interest, i.e., the time window for integrating (observing) the process. These three metrics generate, in tum, a metric which scales the fractional change. Thus, a single and very simple concept provides a master key for analyzing and scaling all transfer processes of interest

All scaling groups which characterize various thermal and fluid dynamic processes may be derived from the fractional change scaling and analysis (FCSA) methodology. This is illustrated by applying the method to three hierarchical levels characterized by three scales: macro, meso and micro.

Thus, at the macro level, the FCSA generates dimensionless groups which scale pressure drop, heat transfer, nuclear reactors and life span of mammals.

At the meso level, the FCSA generates the Mach, Froude and other dimensionless groups when applied to waves and vibrations, and the friction factors for laminar and turbulent flows when applied to diffusion- and vorticity-dominated flows. Furthermore, for turbulent flows, the method can be used to derive the Kolmogorov-5/3 equation, and to demonstrate the wave-eddy duality which is analogous to the waveparticle duality in quantum mechanics.

At the micro level, the FCSA method generates Kolmogorov’s micro-scale parameters for length, frequency and velocity.

Thus, a single concept and a single method can be used to scale and analyze transfer processes associated with particles, waves, diffusion and vorticity.

In closure, it is shown that the equations derived in this paper are analogous to three well-known equations in quantum mechanics, i.e., to the equations of de Broglie, Compton and Planck-Einstein, and that life span of mammals is a manifestation of Noether’s theorem.