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In dimensional analysis, a dimensionless quantity is a quantity without a physical unit and is thus a pure number. Such a number is typically defined as a product or ratio of quantities that have units individually, but cancel out in the combination.

Dimensionless quantities are widely used in the fields of mathematics, physics, engineering, and economics, but also in everyday life.

Contents 1 Properties 2 Buckingham π theorem 2.1 Example 3 Standards efforts 4 Examples 5 List of dimensionless quantities 6 Dimensionless physical constants 7 See also 8 References 9 External links

[edit] 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 (for example to distinguish a mass ratio from a volume ratio).
• 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.

[edit] 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 quantities. For the purposes of the experimenter, different systems which share the same description by dimensionless quantity are equivalent.

[edit] 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 a very important dimensionless number; it describes the fluid flow regime)
• Power number (describes the stirrer and also involves the density of the fluid)

[edit] Standards efforts

The CIPM Consultative Committee for Units contemplated defining the unit of 1 as the 'uno', but the idea was dropped.^[1][2][3][4]

[edit] Examples

Consider this example: Sarah says, "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. Angles are typically measured as the ratio of the length of an arc lying on a circle (with its center being the vertex of the angle) swept out by the angle, compared to some other length. The ratio, length divided by length, is dimensionless. When using the unit radians, the length that is compared is the length of the radius of the circle. When using the unit degree, the length that is compared is 1/360 of the circumference of the circle.

In case of dimensionless quantities the unit is a quotient of like dimensioned quantities that can be reduced to a number (kg/kg = 1, μg/g = 10⁻⁶). Dimensionless quantities can also carry dimensionless units like % (= 0.01), ppt (= 10⁻³), ppm (= 10⁻⁶), ppb (= 10⁻⁹).

[edit] 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	Standard Symbol	Definition	Field of application
Abbe number	V		optics (dispersion in optical materials)
Albedo	α		climatology, astronomy (reflectivity of surfaces or bodies)
Archimedes number	Ar		motion of fluids due to density differences
Arrhenius Number	α		Ratio of activation energy to thermal energy[5]
Atomic weight	M		chemistry
Bagnold number	Ba		flow of bulk solids such as grain and sand. [6]
Bingham Number	Bm		Ratio of yield stress to viscous stress[5]
Biot number	Bi		surface vs. volume conductivity of solids
Bodenstein number			residence-time distribution
Bond number	Bo		capillary action driven by buoyancy [7]
Brinkman number	Br		heat transfer by conduction from the wall to a viscous fluid
Brownell Katz number			combination of capillary number and Bond number
Capillary number	Ca		fluid flow influenced by surface tension
Coefficient of static friction	μs		friction of solid bodies at rest
Coefficient of kinetic friction	μk		friction of solid bodies in translational motion
Colburn j factor			dimensionless heat transfer coefficient
Courant-Friedrich-Levy number	ν		numerical solutions of hyperbolic PDEs [8]
Damkohler number	Da		reaction time scales vs. transport phenomena
Darcy friction factor	Cf or f		fluid flow
Dean number	D		vortices in curved ducts
Deborah number	De		rheology of viscoelastic fluids
Decibel	dB		ratio of two intensities of sound
Drag coefficient	Cd		flow resistance
Euler's number	e		mathematics
Eckert number	Ec		convective heat transfer
Ekman number	Ek		geophysics (frictional (viscous) forces)
Elasticity (economics)	E		widely used to measure how demand or supply responds to price changes
Eötvös number	Eo		determination of bubble/drop shape
Ericksen number	Er		liquid crystal flow behavior
Euler number	Eu		hydrodynamics (pressure forces vs. inertia forces)
Fanning friction factor	f		fluid flow in pipes [9]
Feigenbaum constants	α,δ		chaos theory (period doubling) [10]
Fine structure constant	α		quantum electrodynamics (QED)
f-number	f		optics, photography
Foppl–von Karman number			thin-shell buckling
Fourier number	Fo		heat transfer
Fresnel number	F		slit diffraction [11]
Froude number	Fr		wave and surface behaviour
Gain			electronics (signal output to signal input)
Galilei number	Ga		gravity-driven viscous flow
Golden ratio			mathematics and aesthetics
Graetz number	Gz		heat flow
Grashof number	Gr		free convection
Hatta number	Ha		adsorption enhancement due to chemical reaction
Hagen number	Hg		forced convection
Hydraulic gradient	i		groundwater flow
Karlovitz number			turbulent combustion turbulent combustion
Keulegan–Carpenter number	KC		ratio of drag force to inertia for a bluff object in oscillatory fluid flow
Knudsen number	Kn		continuum approximation in fluids
Kt/V			medicine
Kutateladze number	K		counter-current two-phase flow
Laplace number	La		free convection within immiscible fluids
Lewis number	Le		ratio of mass diffusivity and thermal diffusivity
Lift coefficient	CL		lift available from an airfoil at a given angle of attack
Lockhart-Martinelli parameter	χ		flow of wet gases [12]
Lundquist number	S		ratio of a resistive time to an Alfvén wave crossing time in a plasma
Mach number	M		gas dynamics
Magnetic Reynolds number	Rm		magnetohydrodynamics
Manning roughness coefficient	n		open channel flow (flow driven by gravity) [13]
Marangoni number	Mg		Marangoni flow due to thermal surface tension deviations
Morton number	Mo		determination of bubble/drop shape
Nusselt number	Nu		heat transfer with forced convection
Ohnesorge number	Oh		atomization of liquids, Marangoni flow
Péclet number	Pe		advection–diffusion problems
Peel number			adhesion of microstructures with substrate [14]
pH	pH		chemistry (cologarithm of dissolved H+ ion activity)
Pi	π		mathematics (ratio of a circle's circumference to its diameter)
Poisson's ratio	ν		elasticity (load in transverse and longitudinal direction)
Power factor			electronics (real power to apparent power)
Power number	Np		power consumption by agitators
Prandtl number	Pr		convection heat transfer (thickness of thermal and momentum boundary layers)
Pressure coefficient	CP		pressure experienced at a point on an airfoil
Radian	rad		measurement of angles
Rayleigh number	Ra		buoyancy and viscous forces in free convection
Refractive index	n		electromagnetism, optics
Reynolds number	Re		Ratio of fluid inertial and viscous forces[5]
Relative density	RD		hydrometers, material comparisons
Richardson number	Ri		effect of buoyancy on flow stability [15]
Rockwell scale			mechanical hardness
Rossby number	Ro		inertial forces in geophysics
Rouse number	Z or P		Sediment transport
Schmidt number	Sc		fluid dynamics (mass transfer and diffusion) [16]
Shape factor	H		ratio of displacement thickness to momentum thickness in boundary layer flow
Sherwood number	Sh		mass transfer with forced convection
Sommerfeld number			boundary lubrication [17]
Stanton number	St		heat transfer in forced convection
Stefan number	Ste		heat transfer during phase change
Stokes number	Stk		particle dynamics
Strain	ε		materials science, elasticity
Strouhal number	Sr		continuous and pulsating flow [18]
Taylor number	Ta		rotating fluid flows
Ursell number	U		nonlinearity of surface gravity waves on a shallow fluid layer
van 't Hoff factor	i		quantitative analysis (Kf and Kb)
Wallis parameter	J*		nondimensional superficial velocity in multiphase flows
Weaver flame speed number			laminar burning velocity relative to hydrogen gas [19]
Weber number	We		multiphase flow with strongly curved surfaces
Weissenberg number	Wi		viscoelastic flows [20]
Womersley number	α		continuous and pulsating flows [21]

[edit] Dimensionless physical constants

Certain fundamental physical constants, such as the speed of light in a vacuum, the universal gravitational constant, and the constants of Planck and Boltzmann, 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 natural or 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 constants include:

• α, the fine structure constant, the coupling constant for the electromagnetic interaction;
• μ or β, the proton-to-electron mass ratio, the rest mass of the proton divided by that of the electron. More generally, the rest masses of all elementary particles relative to that of the electron;
• α_s, the coupling constant for the strong force;
• α_G, the gravitational coupling constant.

[edit] See also

• Similitude (model)
• Orders of magnitude (numbers)
• Dimensional analysis
• Normalization (statistics) and standardized moment, the analogous concepts in statistics

[edit] References

[edit] External links

• John Baez, "How Many Fundamental Constants Are There?"
• Huba, J. D., 2007, NRL Plasma Formulary: Dimensionless Numbers of Fluid Mechanics. Naval Research Laboratory. Pp. p. 23, p. 24 and p. 25
• Sheppard, Mike, 2007, "Systematic Search for Expressions of Dimensionless Constants using the NIST database of Physical Constants."
• Biographies of 16 scientists having dimensionless numbers of heat and mass transfer named after them.