Dimensionless quantity

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 the units cancel out.

## Example

_"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.

## 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. http://www2.umt.edu/Geology/faculty/hendrix/g432/g432_L6.htm(http://www2.umt.edu/Geology/faculty/hendrix/g432/g432_L6.htm)|
|Biot number|surface vs. volume conductivity of solids|
|Bodenstein number|residence-time distribution|
|Bond number|capillary action driven by buoyancy http://ising.phys.cwru.edu/plt/PapersInPdf/181BridgeCollapse.pdf(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 kinetic friction|friction of solid bodies in translational motion|
|Colburn j factor|dimensionless heat transfer coefficient|
|Courant-Friedrich-Levy number|numerical solutions of hyperbolic PDEshttp://www.cnrm.meteo.fr/aladin/newsletters/news22/J_Vivoda/Texte.html(http://www.cnrm.meteo.fr/aladin/newsletters/news22/J_Vivoda/Texte.html)|
|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|mathematics|
|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 http://www.engineering.uiowa.edu/~cee081/Exams/Final/Final.htm(http://www.engineering.uiowa.edu/~cee081/Exams/Final/Final.htm)|
|Feigenbaum constants|chaos theory (period doubling) http://www.drchaos.net/drchaos/Book/node44.html(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(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|
|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 http://www.flowprogramme.co.uk/publications/guidancenotes/GN40.pdf(http://www.flowprogramme.co.uk/publications/guidancenotes/GN40.pdf)|
|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 http://web.imech.ac.cn/efile/2000.htm(http://web.imech.ac.cn/efile/2000.htm)|
|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|
|Refractive index|electromagnetism, optics|
|Reynolds number|flow behavior (inertia vs. viscosity)|
|Richardson number|effect of buoyancy on flow stability http://apollo.lsc.vsc.edu/classes/met455/notes/section4/2.html(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(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/(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, elasticity|
|Strouhal number|continuous and pulsating flow http://www.seas.upenn.edu/courses/belab/LabProjects/2001/be310s01m2.doc(http://www.seas.upenn.edu/courses/belab/LabProjects/2001/be310s01m2.doc)|
|Taylor number|rotating fluid flows|
|van 't Hoff factor|quantitative analysis (K<sub>f</sub> and 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(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(http://physics.ucsd.edu/~des/Shear1999.pdf)|
|Womersley number|continuous and pulsating flows http://www.seas.upenn.edu/courses/belab/LabProjects/2001/be310s01m2.doc(http://www.seas.upenn.edu/courses/belab/LabProjects/2001/be310s01m2.doc)|

## Other websites

- Biographies of 16 scientists with dimensionless numbers of heat and mass transfer named after them
- How Many Fundamental Constants Are There? by John Baez
- Michael Sheppard, Systematic Search for Expressions of Dimensionless Constants using the NIST database of Physical Constants, 2007