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Chemical engineering, material science, mechanics (A scale to show the energy needed for detaching two solid particles) [13] [14] Cost of transport: COT = energy efficiency, economics (ratio of energy input to kinetic motion) Damping ratio
One important use is in the analysis of control systems. One of the simplest characteristic units is the doubling time of a system experiencing exponential growth , or conversely the half-life of a system experiencing exponential decay ; a more natural pair of characteristic units is mean age/ mean lifetime , which correspond to base e rather ...
Dimensionless numbers (or characteristic numbers) have an important role in analyzing the behavior of fluids and their flow as well as in other transport phenomena. [1] They include the Reynolds and the Mach numbers, which describe as ratios the relative magnitude of fluid and physical system characteristics, such as density, viscosity, speed of sound, and flow speed.
Dimensionless quantities, or quantities of dimension one, [1] are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. [2] [3] Typically expressed as ratios that align with another system, these quantities do not necessitate explicitly defined units.
This technique can ease the analysis of the problem at hand, and reduce the number of free parameters. Small or large sizes of certain dimensionless parameters indicate the importance of certain terms in the equations for the studied flow. This may provide possibilities to neglect terms in (certain areas of) the considered flow.
Scale analysis rules as follows: Rule1-First step in scale analysis is to define the domain of extent in which we apply scale analysis. Any scale analysis of a flow region that is not uniquely defined is not valid. Rule2-One equation constitutes an equivalence between the scales of two dominant terms appearing in the equation. For example,
To make deviations unitless and facilitate comparisons across different datasets, one can nondimensionalize. One common method involves dividing deviations by a measure of scale( statistical dispersion ), with the population standard deviation used for standardizing or the sample standard deviation for studentizing (e.g., Studentized residual ).
In this article, the following conventions and definitions are to be understood: The Reynolds number Re is taken to be Re = V D / ν, where V is the mean velocity of fluid flow, D is the pipe diameter, and where ν is the kinematic viscosity μ / ρ, with μ the fluid's Dynamic viscosity, and ρ the fluid's density.