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In dimensional analysis, a 'dimensionless quantity' (or more precisely, a 'quantity with the dimensions of 1') is a quantity without any physical units and thus a pure number. 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.

Contents

Examples

Properties

Buckingham π-theorem

Example

List of dimensionless quantities

Dimensionless physical constants

See also

External links

Examples

''"out of every 10 apples I gather, 1 is rotten."'' -- the rotten-to-gathered ratio is (1 apple) / (10 apples) = 0.1 = 10%, 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 quantities are widely used in the fields of mathematics, physics, engineering, and economics but also in everyday life. Whenever one measures 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. A quantity Q is defined as the product of that dimensionless number ''n'' (the number of tick marks) and the unit U (the standard):
:: $mathrm\{Q\}\; stackrel\{mathrm\{def\}\}\{=\}\; n\; mathrm\{U\}$
But, ultimately, people always work with dimensionless numbers in reading measuring instruments and manipulating (changing or calculating with) even dimensionful quantities.
In case of dimensionless quantities the unit U is a quotient of like dimensioned quantities that can be reduced to a number (kg/kg = 1, μg/g = 1e-6). Dimensionless quantities can also carry dimensionless units like % (=0.01), ppm (=1e-6), ppb (=1e-9), ppt (=1e-12).
The CIPM Consultative Committee for Units toyed with the idea of defining the unit of 1 as the 'uno', but the idea was dropped. [1] [2] [3]
[4]

Properties

★ A dimensionless quantity 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 proportion has the same value regardless of the measurement units used to calculate it. It has the same value whether it was calculated using the SI system of units or the imperial system of units. This doesn't hold for all dimensionless quantities; it is guaranteed to hold only for proportions.

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 quantity. For the purposes of the experimenter, different systems which share the same description by dimensionless quantity 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'' (m)

★ Time: ''T'' (s)

★ Mass: ''M'' (kg)
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 quantities

There are infinitely many dimensionless quantities and they are often called numbers. Some of those that are used most often have been given names, as in the following list of examples (alphabetical order):

Name	Field of application
Abbe number	optics (dispersion in optical materials)
Albedo	climatology, astronomy (reflectivity of surfaces or bodies)
Archimedes number	motion of fluids due to density differences
Bagnold number	flow of grain, sand, etc. [5]
Biot number	surface vs. volume conductivity of solids
Bodenstein number	residence-time distribution
Bond number	capillary action driven by buoyancy [6]
Brinkman number	heat transfer by conduction from the wall to a viscous fluid
Brownell Katz number	combination of capillary number and Bond number
Capillary number	fluid flow influenced by surface tension
Coefficient of static friction	friction of solid bodies at rest
Coefficient of kinetic friction	friction of solid bodies in translational motion
Courant-Friedrich-Levy number	non-hydrostatic dynamics [7]
Damköhler numbers	reaction time scales vs. transport phenomena
Darcy friction factor	fluid flow
Dean number	vortices in curved ducts
Deborah number	rheology of viscoelastic fluids
Drag coefficient	flow resistance
Eckert number	convective heat transfer
Ekman number	geophysics (frictional (viscous) forces)
Elasticity (economics)	widely used to measure how demand or supply responds to price changes
Eötvös number	determination of bubble/drop shape
Euler number	hydrodynamics (pressure forces vs. inertia forces)
Fanning friction factor	fluid flow in pipes [8]
Feigenbaum constants	chaos theory (period doubling) [9]
Fine structure constant	quantum electrodynamics (QED)
Foppl–von Karman number	thin-shell buckling
Fourier number	heat transfer
Fresnel number	slit diffraction [10]
Froude number	wave and surface behaviour
Gain	electronics (signal output to signal input)
Galilei number	gravity-driven viscous flow
Graetz number	heat flow
Grashof number	free convection
Hagen number	forced convection
Karlovitz number	turbulent combustion
Knudsen number	continuum approximation in fluids
Kt/V	medicine
Laplace number	free convection within immiscible fluids
Lewis number	ratio of mass diffusivity and thermal diffusivity
Lockhart-Martinelli parameter	flow of wet gases [11]
Lift coefficient	lift available from an airfoil at a given angle of attack
Mach number	gas dynamics
Magnetic Reynolds number	magnetohydrodynamics
Manning roughness coefficient	open channel flow (flow driven by gravity)
Marangoni number	Marangoni flow due to thermal surface tension deviations
Morton number	determination of bubble/drop shape
Nusselt number	heat transfer with forced convection
Ohnesorge number	atomization of liquids, Marangoni flow
Péclet number	advection–diffusion problems
Peel number	adhesion of microstructures with substrate [13]
Pi	mathematics (ratio of a circle's circumference to its diameter)
Poisson's ratio	elasticity (load in transverse and longitudinal direction)
Power factor	electronics (real power to apparent power)
Power number	power consumption by agitators
Prandtl number	forced and free convection
Pressure coefficient	pressure experienced at a point on an airfoil
Radian	measurement of angles
Rayleigh number	buoyancy and viscous forces in free convection
Reynolds number	flow behavior (inertia vs. viscosity)
Richardson number	effect of buoyancy on flow stability [14]
Rockwell scale	mechanical hardness
Rossby number	inertial forces in geophysics
Schmidt number	fluid dynamics (mass transfer and diffusion) [15]
Sherwood number	mass transfer with forced convection
Sommerfeld number	boundary lubrication [16]
Stanton number	heat transfer in forced convection
Stefan number	heat transfer during phase change
Stokes number	particle dynamics
Strain	materials science, elasticity
Strouhal number	continuous and pulsating flow [17]
Taylor number	rotating fluid flows
van 't Hoff factor	quantitative analysis (Kf and Kb)
Weaver flame speed number	laminar burning velocity relative to hydrogen gas [18]
Weber number	multiphase flow with strongly curved surfaces
Weissenberg number	viscoelastic flows [19]
Womersley number	continuous and pulsating flows [20]

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:

★ , the fine structure constant and the electromagnetic coupling constant

★ , 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

See also

★ Similitude (model)

★ Orders of magnitude (numbers)

★ Dimensional analysis

External links

★ - Biographies of 16 scientists with dimensionless numbers of heat and mass transfer named after them

★ How Many Fundamental Constants Are There? by John Baez

★ Systematic Search for Expressions of Dimensionless Constants using the NIST database of Physical Constants