Dimensionless number

In dimensional analysis, a dimensionless number (or more precisely, a number with the dimensions of 1) is a pure number without any physical units. Such a number is typically defined as a product or ratio of quantities which do have units, in such a way that all units cancel.

For example: "one out of every 10 apples I gather is rotten." -- the rotten-to-gathered ratio is (1 apple) / (10 apples) = 0.1, which is a dimensionless quantity. Another more typical example in physics and engineering is the measure of plane angles with the unit of "radian". An angle measured this way is the length of arc lying on a circle (with center being the vertex of the angle) swept out by the angle to the length of the radius of the circle. The units of the ratio is length divided by length which is dimensionless.

Dimensionless numbers are widely used in the fields of mathematics, physics, and engineering but also in everyday life. Whenever one measures anything, any physical quantity, they are measuring that physical quantity against a like dimensioned standard. Whenever one commonly measures a length with a ruler or tape measure, they are counting tick marks on the standard of length they are using, which is a dimensionless number. When they attach that dimensionless number (the number of tick marks) to the units that the standard represents, they conceptually are referring to a dimensionful quantity. But, ultimately, we always work with dimensionless numbers in measuring and manipulating even dimensionful quantities.

The CIPM Consultative Committee for Units toyed with the idea of defining the unit of 1 as the 'uno', but the idea was dropped. ^* ^*

Properties

A dimensionless number has no physical unit associated with it. However, it is sometimes helpful to use the same units in both the numerator and denominator, such as kg/kg, to show the quantity being measured.
A dimensionless number has the same value regardless of the measurement units used to calculate it. It has the same value whether it was calculated using the metric measurement system or the imperial measurement system.
However, a physical quantity may be dimensionless in one system of units and not dimensionless in another system of units. For example, in the nonrationalized cgs system of units, the unit of electric charge (the statcoulomb) is defined in such a way so that the permittivity of free space ε₀ = 1/(4π) whereas in the rationalized SI system, it is ε₀ = 8.85419×10^-12 F/m. In systems of natural units (e.g. Planck units or atomic units), the physical units are defined in such a way that several fundamental constants are made dimensionless and set to 1 (thus removing these scaling factors from equations). While this is convenient in some contexts, abolishing of all or most units and dimensions makes practical physical calculations more error prone.

Buckingham π-theorem

According to the Buckingham π-theorem of dimensional analysis, the functional dependence between a certain number (e.g., n) of variables can be reduced by the number (e.g., k) of independent dimensions occurring in those variables to give a set of p = n − k independent, dimensionless numbers. For the purposes of the experimenter, different systems which share the same description by dimensionless numbers are equivalent.

Example

The power consumption of a stirrer with a particular geometry is a function of the density and the viscosity of the fluid to be stirred, the size of the stirrer given by its diameter, and the speed of the stirrer. Therefore, we have n = 5 variables representing our example.

Those n = 5 variables are built up from k = 3 dimensions which are:

Length L ^*
Time T ^*
Mass M ^*

According to the π-theorem, the n = 5 variables can be reduced by the k = 3 dimensions to form p = n − k = 5 − 3 = 2 independent dimensionless numbers which are in case of the stirrer

Reynolds number (This is the most important dimensionless number; it describes the fluid flow regime)
Power number (describes the stirrer and also involves the density of the fluid)

List of dimensionless numbers

Name
Abbe number
Albedo
Archimedes number
Bagnold number
Biot number
Bodenstein number
Bond number
Brownell Katz number
Capillary number
Coefficient
Coefficient
Cou
Damköhler numbers
Darcy
Deborah number
Drag coefficient
Eckert number
Ekman number
Eötvös number
Euler
Feigenbaum constants
Fourier number
Fresnel number
Froude number
Gain
Galilei number
Graetz number
Grashof number
Hagen number
Knudsen number
Laplace number
Lewis number
Loc
Mach number
Manning
Ohnesorge number
Péclet number
Peel number
Pressure coefficient
Poisson's ratio
Power factor
Power number
Prandtl number
Rayleigh number
Reynolds
Reynolds
Rockwell scale
Rossby number
Schmidt number
Sherwood number
Sommerfeld number
Stanton number
Stokes number
Strouhal number
van 't Hoff factor
Weaver
Weber number
Weissenberg number
Womersley number

are used most often have been given names, as in the following list of examples (alphabetical order): Field of application optics (dispersion in optical materials) climatology, astronomy (reflectivity of surfaces or bodies) motion of fluids due to density differences flow of grain ^* surface vs. volume conductivity of solids residence-time distribution capillary action driven by buoyancy ^* combination of capillary number and Bond number fluid flow influenced by surface tension of static friction of kinetic friction translational motion rant-Friedrich-Levy number href ="http://www.cnrm.meteo.fr/aladin/newsletters/news22/J_Vivoda/Texte.html" target="_blank" >^* reaction time scales vs. transport phenomena friction factor fluid flow rheology of viscoelastic fluids flow resistance convective heat transfer geophysics (frictional (viscous) forces) determination of bubble/drop shape number /hydrodynamics">hydrodynamics (pressure forces vs. inertia forces) friction factor fluid flow in pipes ^* chaos theory (period doubling) ^* heat transfer slit diffraction ^* wave and surface behaviour electronics (signal output to signal input) gravity-driven viscous flow heat flow free convection forced convection continuum approximation in fluids free convection within immiscible fluids ratio of mass diffusivity and thermal diffusivity khart-Martinelli parameter gas">wet gases ^* lift available from an airfoil at a given angle of attack gas dynamics roughness coefficient channel flow">open channel flow (flow driven by gravity) heat transfer with forced convection atomization of liquids viscous vs. Brownian forces adhesion of microstructures with substrate ^* pressure experienced at a point on an airfoil load in transverse and longitudinal direction electronics (real power to apparent power) power consumption by agitators forced and free convection buoyancy and viscous forces in free convection number flow behaviour etohydrodynamics">magnetohydrodynamics ^* effect of buoyancy on flow stability ^* mechanical hardness inertial forces in geophysics fluid dynamics (mass transfer and diffusion) ^* mass transfer with forced convection boundary lubrication ^* heat transfer in forced convection particle dynamics continuous and pulsating flow ^* quantitative analysis (K_f and K_b) flame speed number laminar burning velocity relative to hydrogen gas ^* multiphase flow with strongly curved surfaces viscoelastic flows ^* continuous and pulsating flows ^*

Dimensionless physical constants

Certain physical constants, such as the speed of light in a vacuum, are normalized to 1 if the units for time, length, mass, charge, and temperature are chosen appropriately. The resulting system of units is known as Planck units. However, a handful of dimensionless physical constants cannot be eliminated in any system of units; their values must be determined experimentally. The resulting fundamental physical constants include:

$\alpha$ , the fine structure constant and the electromagnetic coupling constant
$\beta$ , the ratio of the rest mass of the proton to that of the electron
more generally, the masses of all fundamental particles relative to that of the electron
the strong coupling constant
the gravitational coupling constant

External links

http://www.ichmt.org/dimensionless/dimensionless.html - Biographies of 16 scientists with dimensionless numbers of heat and mass transfer named after them
How Many Fundamental Constants Are There? by John Baez

Physical constants | Dimensionless numbers

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