Dimensionless quantity 51331 225820628 2008-07-15T15:38:46Z Mwtoews 711150 /* List of dimensionless quantities */ +Hydraulic gradient {{unreferenced|date=May 2008}} In [[dimensional analysis]], a '''dimensionless quantity''' (or more precisely, a '''quantity with the dimensions of 1''') is a [[quantity]] without any [[physical unit]]s and thus a pure number. Such a number is typically defined as a [[product (mathematics)|product]] or [[ratio]] of [[quantity|quantities]] which do have units, in such a way that all the units cancel out. == Examples == ''"out of every 10 apples I gather, 1 is rotten."'' -- the rotten-to-gathered ratio is (1 rotten apple) / (10 gathered apples) = 0.1 = 10%, which is a dimensionless quantity. Another more typical example in physics and engineering is the measure of [[plane angle]]s. 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 of "[[radians]]" the length that is compared is the length of the radius of the circle. When using the unit of "[[degree (angle)|degrees]]" the length that is compared is 1/360 of the circumference of the circle. 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): :: <math> \mathrm{Q} \ \stackrel{\mathrm{def}}{=}\ n \mathrm{U} \ </math> But, ultimately, people always work with dimensionless numbers in reading [[metrology|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 = 1<sup>-6</sup>). Dimensionless quantities can also carry dimensionless units like % (=0.01), [[Parts-per_notation|ppt]] (=10<sup>-3</sup>), ppm (=10<sup>-6</sup>), ppb (=10<sup>-9</sup>). The [[CIPM]] Consultative Committee for Units toyed with the idea of defining the unit of 1 as the 'uno', but the idea was dropped. [http://www.bipm.fr/utils/common/pdf/CCU15.pdf] [http://www.bipm.fr/utils/common/pdf/CCU16.pdf] [http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=pubmed&dopt=Abstract&list_uids=15588029&query_hl=3] [http://www.iupac.org/publications/ci/2005/2703/bw1_dybkaer.html] == 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 [[variable]]s can be reduced by the number (e.g., ''k'') of [[independent variable|independent]] [[dimension]]s occurring in those variables to give a set of ''p'' = ''n'' &minus; ''k'' independent, dimensionless [[quantity]]. For the purposes of the experimenter, different systems which share the same description by dimensionless [[quantity]] are equivalent. === Example === The [[electric power|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'' &minus; ''k'' = 5 &minus; 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) == List of dimensionless quantities== There are infinitely many dimensionless [[quantity|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): {| class=wikitable ! Name !! Field of application |- | [[Abbe number]] || [[optics]] ([[dispersion (optics)|dispersion]] in optical materials) |- | [[Albedo]] || [[climatology]], [[astronomy]] ([[reflectivity]] of surfaces or bodies) |- | [[Archimedes number]] || motion of [[fluid]]s due to [[density]] differences |- | [[Bagnold number]] || flow of [[grain]], [[sand]], etc. [http://www2.umt.edu/Geology/faculty/hendrix/g432/g432_L6.htm] |- | [[Biot number]] || surface vs. volume [[electrical conductivity|conductivity]] of solids |- | [[Bodenstein number]] || [[residence time|residence-time]] distribution |- | [[Bond number]] || [[capillary action]] driven by [[buoyancy]] [http://ising.phys.cwru.edu/plt/PapersInPdf/181BridgeCollapse.pdf] |- | [[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 friction|Coefficient of kinetic friction]] | friction of solid bodies in translational motion |- | [[Chilton and Colburn J-factor analogy|Colburn j factor]] | dimensionless heat transfer coefficient |- | [[Courant-Friedrich-Levy number]] | numerical solutions of [[hyperbolic PDE]]s[http://www.cnrm.meteo.fr/aladin/newsletters/news22/J_Vivoda/Texte.html] |- | [[Courtin number]] || torque on rotating fluids |- | [[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 |- | [[Decibel]] || ratio of two intensities of sound |- | [[Drag coefficient]] || flow resistance |- | [[E (mathematical constant)|e]]|| [[mathematics]] |- | [[Eckert number]] || convective heat transfer |- | [[Ekman number]] || [[geophysics]] (frictional ([[viscosity|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 (physics)|Euler number]] | [[hydrodynamics]] (pressure forces vs. inertia forces) |- | [[Fanning friction factor]] || fluid flow in pipes [http://www.engineering.uiowa.edu/~cee081/Exams/Final/Final.htm] |- | [[Feigenbaum constants]] || [[chaos theory]] ([[period doubling]]) [http://www.drchaos.net/drchaos/Book/node44.html] |- | [[Fine structure constant]] || [[quantum electrodynamics]] (QED) |- | [[Foppl–von Karman number]] || thin-shell buckling |- | [[Fourier number]] || [[heat]] transfer |- | [[Fresnel number]] || slit [[diffraction]] [http://www.ilt.fraunhofer.de/default.php?web=1&id=100050&lan=eng&dat=2] |- | [[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]] |- | [[Hatta number]] || adsorption enhancement due to chemical reaction |- | [[Hagen number]] || forced convection |- | [[Hydraulic gradient]] || [[groundwater]] flow |- | [[Karlovitz number]] || [[turbulent combustion]] |- | [[Keulegan–Carpenter number]] || ratio of [[drag force]] to [[inertia]] for a bluff object in [[oscillation|oscillatory]] fluid flow |- | [[Knudsen number]] || [[continuum approximation]] in fluids |- | [[Kt/V]] || [[medicine]] |- | [[Laplace number]] || free convection within [[Miscibility|immiscible]] fluids |- | [[Lewis number]] || ratio of mass diffusivity and thermal diffusivity |- | [[Lockhart-Martinelli parameter]] | flow of [[wet gas]]es [http://www.flowprogramme.co.uk/publications/guidancenotes/GN40.pdf] |- | [[Lift coefficient]] || [[Lift (force)|lift]] available from an [[airfoil]] at a given [[angle of attack]] |- | [[Mach number]] || [[gas dynamics]] |- | [[Magnetic Reynolds number]] || [[magnetohydrodynamics]] |- | [[Manning formula|Manning roughness coefficient]] | [[open channel flow]] (flow driven by gravity) {{PDFlink|[http://www.epa.gov/ORD/NRMRL/pubs/600r01043/600R01043chap2.pdf]|109&nbsp;[[Kibibyte|KiB]]<!-- application/pdf, 111618 bytes -->}} |- | [[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]]&ndash;[[diffusion]] problems |- | [[Peel number]] || adhesion of microstructures with substrate [http://web.imech.ac.cn/efile/2000.htm] |- | [[Pi]] || [[mathematics]] (ratio of a circle's circumference to its diameter) |- | [[Poisson's ratio]] || [[Elasticity (physics)|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 |- | [[Refractive index]] || electromagnetism, optics |- | [[Reynolds number]] || flow behavior ([[inertia]] vs. [[viscosity]]) |- | [[Relative density]] || [[hydrometer]]s, material [[comparison]]s |- | [[Richardson number]] || effect of buoyancy on flow stability [http://apollo.lsc.vsc.edu/classes/met455/notes/section4/2.html] |- | [[Rockwell scale]] || mechanical [[hardness]] |- | [[Rossby number]] || inertial forces in [[geophysics]] |- | [[Schmidt number]] || fluid dynamics (mass transfer and [[diffusion]]) [http://www.ent.ohiou.edu/~hbwang/fluidynamics.htm] |- | [[Sherwood number]] || mass transfer with forced convection |- | [[Sommerfeld number]] || boundary [[lubrication]] [http://epubl.luth.se/avslutade/0348-8373/41/] |- | [[Stanton number]] || heat transfer in forced [[convection]] |- | [[Stefan number]] || heat transfer during phase change |- | [[Stokes number]] || particle dynamics |- | [[Strain (materials science)|Strain]] || [[materials science]], [[Elasticity (physics)|elasticity]] |- | [[Strouhal number]] || continuous and pulsating flow [http://www.seas.upenn.edu/courses/belab/LabProjects/2001/be310s01m2.doc] |- | [[Taylor number]] || rotating fluid flows |- | [[van 't Hoff factor]] || [[Quantitative analysis (chemistry)|quantitative analysis]] ([[Freezing-point depression|K<sub>f</sub>]] and [[Boiling point elevation|K<sub>b</sub>]]) |- | [[Weaver flame speed number]] || laminar burning velocity relative to [[hydrogen]] gas [http://eyrie.shef.ac.uk/will/eee/cpe630/comfun8.html] |- | [[Weber number]] || multiphase flow with strongly curved surfaces |- | [[Weissenberg number]] || [[viscoelastic]] flows [http://physics.ucsd.edu/~des/Shear1999.pdf] |- | [[Womersley number]] || continuous and pulsating flows [http://www.seas.upenn.edu/courses/belab/LabProjects/2001/be310s01m2.doc] |} == Dimensionless physical constants == Certain [[physical constant]]s, such as the [[speed of light]] in a vacuum, are normalized to 1 if the units for [[time]], [[length]], [[mass]], [[electric charge|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 constant]]s include: * <math>\alpha</math>, the [[fine structure constant]] and the electromagnetic [[coupling constant]] * <math>\beta</math>, 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]] * [[Normalization (statistics)]] and [[Standardized moment]], the analogous concepts in [[statistics]] ==External links== * [http://www.ichmt.org/dimensionless/dimensionless.html Biographies of 16 scientists with dimensionless numbers of heat and mass transfer named after them] * [http://math.ucr.edu/home/baez/constants.html How Many Fundamental Constants Are There? by John Baez] * [http://www.ipp.mpg.de/~dpc/nrl/ NRL Plasma Formulary], Dimensionless Numbers of Fluid Mechanics, pp. [http://www.ipp.mpg.de/~dpc/nrl/23.html p. 23], [http://www.ipp.mpg.de/~dpc/nrl/24.html p. 24] and [http://www.ipp.mpg.de/~dpc/nrl/25.html p. 25]), J.D. Huba, [[United States Naval Research Laboratory|Naval Research Laboratory]] (2007) * [http://www.msu.edu/~sheppa28/constants/constants.html Systematic Search for Expressions of Dimensionless Constants using the NIST database of Physical Constants] Mike Sheppard, 2007 [[Category:Physical constants]] [[Category:Dimensionless numbers| ]] [[ar:أرقام لابعدية]] [[bs:Bezdimenzionalne veličine]] [[ca:Nombre adimensional]] [[cs:Bezrozměrná veličina]] [[de:Dimensionslose Kennzahl]] [[es:Magnitud adimensional]] [[fr:Grandeur sans dimension]] [[it:Gruppo adimensionale]] [[he:גודל חסר ממד]] [[nl:Dimensieloos getal]] [[ja:無次元数]] [[pl:Liczby podobieństwa]] [[simple:Dimensionless quantity]] [[sl:Brezrazsežna količina]] [[fi:Dimensioton suure]] [[sv:Dimensionslös storhet]] [[zh:无量纲]]