Dimensionless quantity

In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1.^{[1]} Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and in everyday life (such as in counting). Numerous wellknown quantities, such as π, e, and φ, are dimensionless.
Dimensionless quantities are often defined as products or ratios of quantities that are not dimensionless, but whose dimensions cancel out when their powers are multiplied. This is the case, for instance, with the engineering strain, a measure of deformation. It is defined as change in length over initial length but, since these quantities both have dimensions L (length), the result is a dimensionless quantity.
Contents
Properties
 Even though a dimensionless quantity has no physical dimension associated with it, it can still have dimensionless units (i.e. not unitless). To show the quantity being measured (for example mass fraction or mole fraction), it is sometimes helpful to use the same units in both the numerator and denominator (kg/kg or mol/mol). The quantity may also be given as a ratio of two different units that have the same dimension (for instance, light years over meters). This may be the case when calculating slopes in graphs, or when making unit conversions. Such notation does not indicate the presence of physical dimensions, and is purely a notational convention. Other common dimensionless units are % (= 0.01), ‰ (= 0.001), ppm (= 10^{−6}), ppb (= 10^{−9}), ppt (= 10^{−12}) and angle units (degrees, radians, grad). Units of amount such as the dozen and the gross are also dimensionless.
 The ratio of two quantities with the same dimensions is dimensionless, and has the same value regardless of the units used to calculate them. For instance, if body A exerts a force of magnitude F on body B, and B exerts a force of magnitude f on A, then the ratio F/f will always be equal to 1, regardless of the actual units used to measure F and f. This is a fundamental property of dimensionless proportions and follows from the assumption that the laws of physics are independent of the system of units used in their expression. In this case, if the ratio F/f was not always equal to 1, but changed if we switched from SI to CGS, for instance, that would mean that Newton's Third Law's truth or falsity would depend on the system of units used, which would contradict this fundamental hypothesis. The assumption that the laws of physics are not contingent upon a specific unit system is also closely related to the Buckingham π theorem. A formulation of this theorem is that any physical law can be expressed as an identity (always true equation) involving only dimensionless combinations (ratios or products) of the variables linked by the law (e. g., pressure and volume are linked by Boyle's Law – they are inversely proportional). If the dimensionless combinations' values changed with the systems of units, then the equation would not be an identity, and Buckingham's theorem would not hold.
Buckingham π theorem
Another consequence of the Buckingham π theorem of dimensional analysis is that the functional dependence between a certain number (say, n) of variables can be reduced by the number (say, 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.
Example
The power consumption of a stirrer with a given shape 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 πtheorem, 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 (a dimensionless number describing the fluid flow regime)
 Power number (describing the stirrer and also involves the density of the fluid)
Standards efforts
The International Committee for Weights and Measures contemplated defining the unit of 1 as the 'uno', but the idea was dropped.^{[2]}^{[3]}^{[4]}
Examples
 Consider this example: Sarah says, "Out of every 10 apples I gather, 1 is rotten.". The rottentogathered 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. An angle is measured as the ratio of the length of a circle's arc subtended by an angle whose vertex is the centre of the circle to some other length. The ratio, length divided by length, is dimensionless. When using radians as the unit, the length that is compared is the length of the radius of the circle. When using degree as the units, the arc's length is compared to 1/360 of the circumference of the circle.
 In the case of the dimensionless quantity π, being the ratio of a circle's circumference to its diameter, the number would be constant regardless of what unit is used to measure a circle's circumference and diameter (eg. centimetres, miles, lightyears, etc), as long as the same unit is used for both.
List of dimensionless quantities
All numbers are dimensionless quantities. Certain dimensionless quantities of some importance are given below:
Name Standard symbol Definition Field of application Abbe number V optics (dispersion in optical materials) Activity coefficient γ chemistry (Proportion of "active" molecules or atoms) 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]} Blowdown circulation number BC deviation from isothermal flow in blowdown (rapid depressurization) of a pressure vessel^{[7]} Bejan number
(thermodynamics)Be the ratio of heat transfer irreversibility to total irreversibility due to heat transfer and fluid friction^{[8]} Bejan number
(fluid mechanics)Be dimensionless pressure drop along a channel^{[9]} Bingham number Bm Ratio of yield stress to viscous stress^{[5]} Bingham capillary number Bm.Ca Ratio of yield stress to capillary pressure^{[10]} Biot number Bi surface vs. volume conductivity of solids Blake number Bl or B relative importance of inertia compared to viscous forces in fluid flow through porous media Bodenstein number Bo residencetime distribution Bond number Bo capillary action driven by buoyancy ^{[11]} 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 μ_{s} friction of solid bodies at rest Coefficient of kinetic friction μ_{k} friction of solid bodies in translational motion Colburn j factor dimensionless heat transfer coefficient Courant–Friedrich–Levy number ν numerical solutions of hyperbolic PDEs ^{[12]} Damkohler number Da Da = kτ reaction time scales vs. resonance time Damping ratio ζ the level of damping in a system Darcy friction factor C_{f} or f fluid flow Dean number D vortices in curved ducts Deborah number De rheology of viscoelastic fluids Decibel dB ratio of two intensities, often sound Drag coefficient C_{d} flow resistance Dukhin number Du ratio of electric surface conductivity to the electric bulk conductivity in heterogeneous systems 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 ^{[13]} Feigenbaum constants α,δ chaos theory (period doubling) ^{[14]} Fine structure constant α quantum electrodynamics (QED) fnumber f optics, photography Foppl–von Karman number thinshell buckling Fourier number Fo heat transfer Fresnel number F slit diffraction ^{[15]} Froude number Fr wave and surface behaviour Gain electronics (signal output to signal input) Galilei number Ga gravitydriven viscous flow Golden ratio φ mathematics and aesthetics Graetz number Gz heat flow Grashof number Gr free convection Gravitational coupling constant α_{G} Gravitation Hatta number Ha adsorption enhancement due to chemical reaction Hagen number Hg forced convection Hydraulic gradient i groundwater flow Jakob Number Ja Ratio of sensible to latent energy absorbed during liquidvapor phase change^{[16]} Karlovitz number turbulent combustion turbulent combustion Keulegan–Carpenter number K_{C} ratio of drag force to inertia for a bluff object in oscillatory fluid flow Knudsen number Kn ratio of the molecular mean free path length to a representative physical length scale Kt/V medicine Kutateladze number K countercurrent twophase flow Laplace number La free convection within immiscible fluids Lewis number Le ratio of mass diffusivity and thermal diffusivity Lift coefficient C_{L} lift available from an airfoil at a given angle of attack Lockhart–Martinelli parameter χ flow of wet gases ^{[17]} Love number measuring the solidity of the earth Lundquist number S ratio of a resistive time to an Alfvén wave crossing time in a plasma Mach number M Ratio of current speed to the speed of sound, i.e. Mach 1 is the speed of sound, Mach 0.5 is half the speed of sound, Mach 2 is twice the speed of sound. gas dynamics Magnetic Reynolds number R_{m} magnetohydrodynamics Manning roughness coefficient n open channel flow (flow driven by gravity) ^{[18]} Marangoni number Mg Marangoni flow due to thermal surface tension deviations Morton number Mo determination of bubble/drop shape Mpemba number K_{M} thermal conduction and diffusion in freezing of a solution^{[19]} Nusselt number Nu heat transfer with forced convection Ohnesorge number Oh atomization of liquids, Marangoni flow Péclet number Pe advection–diffusion problems; relates total momentun transfer to molecular heat transfer. Peel number adhesion of microstructures with substrate ^{[20]} Perveance K measure of the strength of space charge in a charged particle beam Pi π mathematics (ratio of a circle's circumference to its diameter) Poisson's ratio ν elasticity (load in transverse and longitudinal direction) Porosity ϕ geology Power factor electronics (real power to apparent power) Power number N_{p} power consumption by agitators Prandtl number Pr convection heat transfer (thickness of thermal and momentum boundary layers) Pressure coefficient C_{P} pressure experienced at a point on an airfoil Q factor Q describes how underdamped an oscillator or resonator is 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 ^{[21]} Rockwell scale mechanical hardness Rolling resistance coefficient C_{rr} vehicle dynamics Rossby number R_{o} inertial forces in geophysics Rouse number Z or P sediment transport Schmidt number Sc fluid dynamics (mass transfer and diffusion) ^{[22]} Shape factor H ratio of displacement thickness to momentum thickness in boundary layer flow Sherwood number Sh mass transfer with forced convection Shields parameter τ_{∗} or θ threshold of sediment movement due to fluid motion Sommerfeld number boundary lubrication ^{[23]} Stanton number St heat transfer in forced convection Stefan number Ste heat transfer during phase change Stokes number Stk or S_{k} particle dynamics in a fluid stream Strain materials science, elasticity Strouhal number St or Sr nondimensional frequency, continuous and pulsating flow ^{[24]} Taylor number Ta rotating fluid flows Ursell number U nonlinearity of surface gravity waves on a shallow fluid layer Vadasz number Va governs the effects of porosity ϕ, the Prandtl number and the Darcy number on flow in a porous medium van 't Hoff factor i quantitative analysis (K_{f} and K_{b}) Wallis parameter J^{*} nondimensional superficial velocity in multiphase flows Weaver flame speed number laminar burning velocity relative to hydrogen gas ^{[25]} Weber number We multiphase flow with strongly curved surfaces Weissenberg number Wi viscoelastic flows ^{[26]} Womersley number α continuous and pulsating flows ^{[27]} 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. However, not all physical constants can be eliminated in any system of units; the values of the remaining ones must be determined experimentally. Resulting constants include:
 α, the fine structure constant, the coupling constant for the electromagnetic interaction;
 μ or β, the protontoelectron 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.
See also
 Similitude (model)
 Orders of magnitude (numbers)
 Dimensional analysis
 Normalization (statistics) and standardized moment, the analogous concepts in statistics
References
 ^ "1.8 (1.6) quantity of dimension one dimensionless quantity". International vocabulary of metrology — Basic and general concepts and associated terms (VIM). ISO. 2008. http://www.iso.org/sites/JCGM/VIM/JCGM_200e_FILES/MAIN_JCGM_200e/01_e.html#L_1_8. Retrieved 20110322.
 ^ "BIPM Consultative Committee for Units (CCU), 15th Meeting" (PDF). 17–18 April 2003. http://www.bipm.fr/utils/common/pdf/CCU15.pdf. Retrieved 20100122.
 ^ "BIPM Consultative Committee for Units (CCU), 16th Meeting" (PDF). http://www.bipm.fr/utils/common/pdf/CCU16.pdf. Retrieved 20100122.
 ^ Dybkaer, René (2004). "An ontology on property for physical, chemical, and biological systems". APMIS Suppl. (117): 1–210. PMID 15588029. http://www.iupac.org/publications/ci/2005/2703/bw1_dybkaer.html.
 ^ ^{a} ^{b} ^{c} "Table of Dimensionless Numbers" (PDF). http://www.cchem.berkeley.edu/gsac/grad_info/prelims/binders/dimensionless_numbers.pdf. Retrieved 20091105.
 ^ Bagnold number
 ^ Katz J. I. (2009). "Circulation in blowdown flows". J. Pressure Vessel Technology 131 (3): 034501. doi:10.1115/1.3110038.
 ^ Paoletti S., Rispoli F., Sciubba E. (1989). "Calculation of exergetic losses in compact heat exchanger passager". ASME AES 10 (2): 21–9.
 ^ Bhattacharjee S., Grosshandler W.L. (1988). "The formation of wall jet near a high temperature wall under microgravity environment". ASME MTD 96: 711–6.
 ^ German G., Bertola V. (2010). "The spreading behaviour of capillary driven yieldstress drops". Colloid Surface A 366: 18–26. doi:10.1016/j.colsurfa.2010.05.019.
 ^ Bond number
 ^ Courant–Friedrich–Levy number
 ^ Fanning friction factor
 ^ Feigenbaum constants
 ^ Fresnel number
 ^ {{cite textbook  Incropera, Frank P., Fundamentals of heat and mass transfer, pg. 376, 2007, John Wiley & Sons, Inc
 ^ Lockhart–Martinelli parameter
 ^ Manning coefficientPDF (109 KB)
 ^ Katz J. I. (2009). "When hot water freezes before cold". Am. J. Phys. 77: 27–29. Bibcode 2009AmJPh..77...27K. doi:10.1119/1.2996187. [1] Mpemba number
 ^ Peel number
 ^ Richardson number
 ^ Schmidt number
 ^ Sommerfeld number
 ^ Strouhal number
 ^ Weaver flame speed number
 ^ Weissenberg number
 ^ Womersley number
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. p. 23, 24, 25
 Sheppard, Mike, 2007, "Systematic Search for Expressions of Dimensionless Constants using the NIST database of Physical Constants."
Categories: Physical constants
 Dimensionless numbers
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