Examine individual changes

'{{unreferenced|date=May 2008}} In [[dimensional analysis]], a '''dimensionless quantity''' refers to 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. Dimensionless quantities are widely used in the fields of [[mathematics]], [[physics]], [[engineering]], and [[economics]] but also in everyday life. ==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 (mathematics)|variables]] 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|quantities]]. 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) ==Standards efforts== The [[CIPM]] Consultative Committee for Units contemplated defining the unit of 1 as the 'uno', but the idea was dropped.<ref>{{cite web|url=http://www.bipm.fr/utils/common/pdf/CCU15.pdf|title=BIPM Consultative Committee for Units (CCU), 15th Meeting|date=17-18 April 2003}}</ref><ref>{{cite web|url=http://www.bipm.fr/utils/common/pdf/CCU16.pdf|title=BIPM Consultative Committee for Units (CCU), 16th Meeting}}</ref><ref>{{cite web|url=http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=pubmed&dopt=Abstract&list_uids=15588029&query_hl=3|title=An ontology on property for physical, chemical, and biological systems.}}</ref><ref> {{cite web|url=http://www.iupac.org/publications/ci/2005/2703/bw1_dybkaer.html|title=An ontology on property for physical, chemical, and biological systems|author=René Dybkaer}}</ref> ==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 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 [[radians]], the length that is compared is the length of the radius of the circle. When using the unit [[degree (angle)|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), [[Parts-per notation|ppt]] (=10^-3), ppm (=10^-6), ppb (=10^-9). ==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 !! Standard Symbol !! Field of application |- | [[Abbe number]] || V || [[optics]] ([[dispersion (optics)|dispersion]] in optical materials) |- | [[Albedo]] || <math>\alpha</math> || [[climatology]], [[astronomy]] ([[reflectivity]] of surfaces or bodies) |- | [[Archimedes number]] || Ar || motion of [[fluid]]s due to [[density]] differences |- | [[Atomic weight]] || M || [[chemistry]] |- | [[Bagnold number]] || Ba || flow of bulk solids such as [[grain]] and [[sand]]. <ref>[http://www2.umt.edu/Geology/faculty/hendrix/g432/g432_L6.htm Bagnold number]</ref> |- | [[Biot number]] || Bi || surface vs. volume [[electrical conductivity|conductivity]] of solids |- | [[Bodenstein number]] || || [[residence time|residence-time]] distribution |- | [[Bond number]] || Bo || [[capillary action]] driven by [[buoyancy]] <ref>[http://ising.phys.cwru.edu/plt/PapersInPdf/181BridgeCollapse.pdf Bond number]</ref> |- | [[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]] || <math>\mu_s</math> || friction of solid bodies at rest |- | [[Coefficient of friction|Coefficient of kinetic friction]] || <math>\mu_k</math> || friction of solid bodies in translational motion |- | [[Chilton and Colburn J-factor analogy|Colburn j factor]] || || dimensionless heat transfer coefficient |- | [[Courant-Friedrich-Levy number]] || <math>\nu</math> || numerical solutions of [[hyperbolic PDE]]s <ref>[http://www.cnrm.meteo.fr/aladin/newsletters/news22/J_Vivoda/Texte.html Courant-Friedrich-Levy number]</ref> |- | [[Damkohler number]] || Da || reaction time scales vs. transport phenomena |- | [[Darcy friction factor]] || <math>C_f</math> or <math>f</math> || 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]] || <math>C_d</math> || flow resistance |- | [[Euler's number]] || e || [[mathematics]] |- | [[Eckert number]] || Ec || convective heat transfer |- | [[Ekman number]] || Ek || [[geophysics]] (frictional ([[viscosity|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 (physics)|Euler number]] || Eu || [[hydrodynamics]] (pressure forces vs. inertia forces) |- | [[Fanning friction factor]] || f || fluid flow in pipes <ref>[http://www.engineering.uiowa.edu/~cee081/Exams/Final/Final.htm Fanning friction factor]</ref> |- | [[Feigenbaum constants]] || <math>\alpha, \delta</math> || [[chaos theory]] ([[period doubling]]) <ref>[http://www.drchaos.net/drchaos/Book/node44.html Feigenbaum constants]</ref> |- | [[Fine structure constant]] || <math>\alpha</math> || [[quantum electrodynamics]] (QED) |- | [[f-number]] || <math>f</math> || [[optics]], [[photography]] |- | [[Foppl–von Karman number]] || || thin-shell buckling |- | [[Fourier number]] || Fo || [[heat]] transfer |- | [[Fresnel number]] || F || slit [[diffraction]] <ref>[http://www.ilt.fraunhofer.de/default.php?web=1&id=100050&lan=eng&dat=2 Fresnel number]</ref> |- | [[Froude number]] || Fr || [[wave]] and surface behaviour |- | [[Gain]] || || [[electronics]] (signal output to signal input) |- | [[Galilei number]] || Ga || gravity-driven viscous flow |- | [[Golden ratio]] || <math>\varphi</math> || [[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]] |- | [[Keulegan–Carpenter number]] || <math>K_C</math> || ratio of [[drag force]] to [[inertia]] for a bluff object in [[oscillation|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 [[Miscibility|immiscible]] fluids |- | [[Lewis number]] || Le || ratio of mass diffusivity and thermal diffusivity |- | [[Lift coefficient]] || <math>C_L</math> || [[Lift (force)|lift]] available from an [[airfoil]] at a given [[angle of attack]] |- | [[Lockhart-Martinelli parameter]] || <math>\chi</math> || flow of [[wet gas]]es <ref>[http://www.flowprogramme.co.uk/publications/guidancenotes/GN40.pdf Lockhart-Martinelli parameter]</ref> |- | [[Lundquist number]] || <math>S</math> || ratio of a resistive time to an [[Alfven wave|Alfvén wave]] crossing time in a plasma |- | [[Mach number]] || M || [[gas dynamics]] |- | [[Magnetic Reynolds number]] || <math>R_m</math> || [[magnetohydrodynamics]] |- | [[Manning formula|Manning roughness coefficient]] || n || [[open channel flow]] (flow driven by gravity) <ref>{{PDFlink|[http://www.epa.gov/ORD/NRMRL/pubs/600r01043/600R01043chap2.pdf Manning coefficient]|109&nbsp;[[Kibibyte|KiB]]<!-- application/pdf, 111618 bytes -->}}</ref> |- | [[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]]&ndash;[[diffusion]] problems |- | [[Peel number]] || || adhesion of microstructures with substrate <ref>[http://web.imech.ac.cn/efile/2000.htm Peel number]</ref> |- | [[Pi]] || <math>\pi</math> || [[mathematics]] (ratio of a circle's circumference to its diameter) |- | [[pH]] || <math>\pH</math> || [[chemistry]] (cologarithm of h+ ion activity) |- | [[Poisson's ratio]] || <math>\nu</math> || [[Elasticity (physics)|elasticity]] (load in transverse and longitudinal direction) |- | [[Power factor]] || || [[electronics]] (real power to apparent power) |- | [[Power number]] || <math>N_p</math> || power consumption by agitators |- | [[Prandtl number]] || Pr || [[convection]] [[heat transfer]] (thickness of thermal and momentum [[boundary layers]]) |- | [[Pressure coefficient]] || <math>C_P</math> || 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 || flow behavior ([[inertia]] vs. [[viscosity]]) |- | [[Relative density]] || RD || [[hydrometer]]s, material [[comparison]]s |- | [[Richardson number]] || Ri || effect of buoyancy on flow stability <ref>[http://apollo.lsc.vsc.edu/classes/met455/notes/section4/2.html Richardson number]</ref> |- | [[Rockwell scale]] || || mechanical [[hardness]] |- | [[Rossby number]] || <math>R_o</math> || inertial forces in [[geophysics]] |- | [[Rouse number]] || Z or P || [[Sediment transport]] |- | [[Schmidt number]] || Sc || fluid dynamics (mass transfer and [[diffusion]]) <ref>[http://www.ent.ohiou.edu/~hbwang/fluidynamics.htm Schmidt number]</ref> |- | [[Shape factor (boundary layer flow)|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]] <ref>[http://epubl.luth.se/avslutade/0348-8373/41/ Sommerfeld number]</ref> |- | [[Stanton number]] || St || heat transfer in forced [[convection]] |- | [[Stefan number]] || Ste || heat transfer during phase change |- | [[Stokes number]] || Stk || particle dynamics |- | [[Strain (materials science)|Strain]] || <math>\epsilon</math> || [[materials science]], [[Elasticity (physics)|elasticity]] |- | [[Strouhal number]] || Sr || continuous and pulsating flow <ref>[http://www.seas.upenn.edu/courses/belab/LabProjects/2001/be310s01m2.doc Strouhal number]</ref> |- | [[Taylor number]] || Ta || rotating fluid flows |- | [[Ursell number]] || U || nonlinearity of [[ocean surface wave|surface gravity waves]] on a shallow fluid layer |- | [[van 't Hoff factor]] || i || [[Quantitative analysis (chemistry)|quantitative analysis]] ([[Freezing-point depression|K_f]] and [[Boiling point elevation|K_b]]) |- | [[Wallis parameter]] || J* || nondimensional superficial velocity in multiphase flows |- | [[Weaver flame speed number]] || || laminar burning velocity relative to [[hydrogen]] gas <ref>[http://eyrie.shef.ac.uk/will/eee/cpe630/comfun8.html Weaver flame speed number]</ref> |- | [[Weber number]] || We || multiphase flow with strongly curved surfaces |- | [[Weissenberg number]] || Wi || [[viscoelastic]] flows <ref>[http://physics.ucsd.edu/~des/Shear1999.pdf Weissenberg number]</ref> |- | [[Womersley number]] || <math>\alpha</math> || continuous and pulsating flows <ref>[http://www.seas.upenn.edu/courses/belab/LabProjects/2001/be310s01m2.doc Womersley number]</ref> |} ==Dimensionless physical constants== Certain [[fundamental physical constant]]s, such as the [[speed of light]] in a vacuum, the [[universal gravitational constant]], and the constants of [[Planck's constant|Planck]] and [[Boltzmann's constant|Boltzmann]], 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 [[natural units|natural]] or [[Planck units]]. However, a handful of [[dimensionless physical constant]]s 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 mass]]es 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== {{reflist}} ==External links== * [[John Baez]], "[http://math.ucr.edu/home/baez/constants.html How Many Fundamental Constants Are There?]" * Huba, J. D., 2007, ''[http://www.ipp.mpg.de/~dpc/nrl/ NRL Plasma Formulary: Dimensionless Numbers of Fluid Mechanics.]'' [[United States Naval Research Laboratory|Naval Research Laboratory]]. 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] * Sheppard, Mike, 2007, "[http://www.msu.edu/~sheppa28/constants/constants.html Systematic Search for Expressions of Dimensionless Constants using the NIST database of Physical Constants.]" * [http://www.ichmt.org/dimensionless/dimensionless.html Biographies] of 16 scientists having dimensionless numbers of heat and mass transfer named after them. 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