List of dimensionless quantities

This is a list of well-known dimensionless quantities illustrating their variety of forms and applications. The tables also include pure numbers, dimensionless ratios, or dimensionless physical constants; these topics are discussed in the article.

Biology and medicine

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Name Standard symbol Definition Field of application
Basic reproduction number   number of infections caused on average by an infectious individual over entire infectious period epidemiology
Body fat percentage total mass of fat divided by total body mass, multiplied by 100 biology
Kt/V Kt/V medicine (hemodialysis and peritoneal dialysis treatment; dimensionless time)
Waist–hip ratio waist circumference divided by hip circumference biology
Waist-to-chest ratio waist circumference divided by chest circumference biology
Waist-to-height ratio waist circumference divided by height biology

Chemistry

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Name Standard symbol Definition Field of application
Activity coefficient     chemistry (Proportion of "active" molecules or atoms)
Arrhenius number     chemistry (ratio of activation energy to thermal energy)[1]
Atomic weight M chemistry (mass of one atom divided by the atomic mass constant, Da)
Bodenstein number Bo or Bd   chemistry (residence-time distribution; similar to the axial mass transfer Peclet number)[2]
Damkohler number Da   chemistry (reaction time scales vs. residence time)
Hatta number Ha   chemical engineering (adsorption enhancement due to chemical reaction)
Jakob number Ja   chemistry (ratio of sensible to latent energy absorbed during liquid-vapor phase change)[3]
pH     chemistry (the measure of the acidity or basicity of an aqueous solution)
van 't Hoff factor i   quantitative analysis (Kf and Kb)
Wagner number Wa   electrochemistry (ratio of kinetic polarization resistance to solution ohmic resistance in an electrochemical cell)[4]
Weaver flame speed number Wea   combustion (laminar burning velocity relative to hydrogen gas)[5]

Physics

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Physical constants

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Fluids and heat transfer

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Name Standard symbol Definition Field of application
Archimedes number Ar   fluid mechanics (motion of fluids due to density differences)
Asakuma number As   heat transfer (ratio of heat generation of microwave dielectric heating to thermal diffusion[6]
Atwood number A   fluid mechanics (onset of instabilities in fluid mixtures due to density differences)
Bagnold number Ba   fluid mechanics, geology (ratio of grain collision stresses to viscous fluid stresses in flow of a granular material such as grain and sand)[7]
Bejan number
(fluid mechanics)
Be   fluid mechanics (dimensionless pressure drop along a channel)[8]
Bejan number
(thermodynamics)
Be   thermodynamics (ratio of heat transfer irreversibility to total irreversibility due to heat transfer and fluid friction)[9]
Bingham number Bm   fluid mechanics, rheology (ratio of yield stress to viscous stress)[1]
Biot number Bi   heat transfer (surface vs. volume conductivity of solids)
Blake number Bl or B   geology, fluid mechanics, porous media (inertial over viscous forces in fluid flow through porous media)
Bond number Bo   geology, fluid mechanics, porous media (buoyant versus capillary forces, similar to the Eötvös number) [10]
Brinkman number Br   heat transfer, fluid mechanics (conduction from a wall to a viscous fluid)
Brownell–Katz number NBK   fluid mechanics (combination of capillary number and Bond number) [11]
Capillary number Ca   porous media, fluid mechanics (viscous forces versus surface tension)
Chandrasekhar number Q   magnetohydrodynamics (ratio of the Lorentz force to the viscosity in magnetic convection)
Colburn J factors JM, JH, JD turbulence; heat, mass, and momentum transfer (dimensionless transfer coefficients)
Darcy friction factor Cf or fD fluid mechanics (fraction of pressure losses due to friction in a pipe; four times the Fanning friction factor)
Dean number D   turbulent flow (vortices in curved ducts)
Deborah number De   rheology (viscoelastic fluids)
Drag coefficient cd   aeronautics, fluid dynamics (resistance to fluid motion)
Eckert number Ec   convective heat transfer (characterizes dissipation of energy; ratio of kinetic energy to enthalpy)
Ekman number Ek   geophysics (viscous versus Coriolis forces)
Eötvös number Eo   fluid mechanics (shape of bubbles or drops)
Ericksen number Er   fluid dynamics (liquid crystal flow behavior; viscous over elastic forces)
Euler number Eu   hydrodynamics (stream pressure versus inertia forces)
Excess temperature coefficient     heat transfer, fluid dynamics (change in internal energy versus kinetic energy)[12]
Fanning friction factor f fluid mechanics (fraction of pressure losses due to friction in a pipe; 1/4th the Darcy friction factor)[13]
Fourier number Fo   heat transfer, mass transfer (ratio of diffusive rate versus storage rate)
Froude number Fr   fluid mechanics (wave and surface behaviour; ratio of a body's inertia to gravitational forces)
Galilei number Ga   fluid mechanics (gravitational over viscous forces)
Görtler number G   fluid dynamics (boundary layer flow along a concave wall)
Graetz number Gz   heat transfer, fluid mechanics (laminar flow through a conduit; also used in mass transfer)
Grashof number Gr   heat transfer, natural convection (ratio of the buoyancy to viscous force)
Hagen number Hg   heat transfer (ratio of the buoyancy to viscous force in forced convection)
Hydraulic gradient i   fluid mechanics, groundwater flow (pressure head over distance)
Karlovitz number Ka   turbulent combustion (characteristic chemical time scale to Kolmogorov time scale)
Keulegan–Carpenter number KC   fluid dynamics (ratio of drag force to inertia for a bluff object in oscillatory fluid flow)
Knudsen number Kn   gas dynamics (ratio of the molecular mean free path length to a representative physical length scale)
Kutateladze number Ku   fluid mechanics (counter-current two-phase flow)[14]
Laplace number La   fluid dynamics (free convection within immiscible fluids; ratio of surface tension to momentum-transport)
Lewis number Le   heat and mass transfer (ratio of thermal to mass diffusivity)
Lift coefficient CL   aerodynamics (lift available from an airfoil at a given angle of attack)
Lockhart–Martinelli parameter     two-phase flow (flow of wet gases; liquid fraction)[15]
Mach number M or Ma   gas dynamics (compressible flow; dimensionless velocity)
Magnetic Reynolds number Rm   magnetohydrodynamics (ratio of magnetic advection to magnetic diffusion)
Manning roughness coefficient n open channel flow (flow driven by gravity)[16]
Marangoni number Mg   fluid mechanics (Marangoni flow; thermal surface tension forces over viscous forces)
Markstein number     fluid dynamics, combustion (turbulent combustion flames)
Morton number Mo   fluid dynamics (determination of bubble/drop shape)
Nusselt number Nu   heat transfer (forced convection; ratio of convective to conductive heat transfer)
Ohnesorge number Oh   fluid dynamics (atomization of liquids, Marangoni flow)
Péclet number Pe   heat transfer (advectiondiffusion problems; total momentum transfer to molecular heat transfer)
Péclet number Pe   mass transfer (advectiondiffusion problems; total momentum transfer to diffusive mass transfer)
Prandtl number Pr   heat transfer (ratio of viscous diffusion rate over thermal diffusion rate)
Pressure coefficient CP   aerodynamics, hydrodynamics (pressure experienced at a point on an airfoil; dimensionless pressure variable)
Rayleigh number Ra   heat transfer (buoyancy versus viscous forces in free convection)
Reynolds number Re   fluid mechanics (ratio of fluid inertial and viscous forces)[1]
Richardson number Ri   fluid dynamics (effect of buoyancy on flow stability; ratio of potential over kinetic energy)[17]
Roshko number Ro   fluid dynamics (oscillating flow, vortex shedding)
Schmidt number Sc   mass transfer (viscous over molecular diffusion rate)[18]
Shape factor H   boundary layer flow (ratio of displacement thickness to momentum thickness)
Sherwood number Sh   mass transfer (forced convection; ratio of convective to diffusive mass transport)
Sommerfeld number S   hydrodynamic lubrication (boundary lubrication)[19]
Stanton number St   heat transfer and fluid dynamics (forced convection)
Stokes number Stk or Sk   particles suspensions (ratio of characteristic time of particle to time of flow)
Strouhal number St or Sr   fluid dynamics (continuous and pulsating flow; nondimensional frequency)[20]
Stuart number N   magnetohydrodynamics (ratio of electromagnetic to inertial forces)
Taylor number Ta   fluid dynamics (rotating fluid flows; inertial forces due to rotation of a fluid versus viscous forces)
Ursell number U   wave mechanics (nonlinearity of surface gravity waves on a shallow fluid layer)
Vadasz number Va   porous media (governs the effects of porosity  , the Prandtl number and the Darcy number on flow in a porous medium) [21]
Wallis parameter j*   multiphase flows (nondimensional superficial velocity)[22]
Weber number We   multiphase flow (strongly curved surfaces; ratio of inertia to surface tension)
Weissenberg number Wi   viscoelastic flows (shear rate times the relaxation time)[23]
Womersley number     biofluid mechanics (continuous and pulsating flows; ratio of pulsatile flow frequency to viscous effects)[24]
Zel'dovich number     fluid dynamics, Combustion (Measure of activation energy)

Solids

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Name Standard symbol Definition Field of application
Coefficient of kinetic friction   mechanics (friction of solid bodies in translational motion)
Coefficient of static friction   mechanics (friction of solid bodies at rest)
Dieterich-Ruina-Rice number     mechanics, friction, rheology, geophysics (stiffness ratio for frictional contacts)[25]
Föppl–von Kármán number     virology, solid mechanics (thin-shell buckling)
Rockwell scale mechanical hardness (indentation hardness of a material)
Rolling resistance coefficient Crr   vehicle dynamics (ratio of force needed for motion of a wheel over the normal force)

Optics

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Name Standard symbol Definition Field of application
Abbe number V   optics (dispersion in optical materials)
f-number N   optics, photography (ratio of focal length to diameter of aperture)
Fresnel number F   optics (slit diffraction)[26]
Refractive index n   electromagnetism, optics (speed of light in vacuum over speed of light in a material)
Transmittance T   optics, spectroscopy (the ratio of the intensities of radiation exiting through and incident on a sample)

Mathematics and statistics

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Name Standard symbol Definition Field of application
Coefficient of determination   statistics (proportion of variance explained by a statistical model)
Coefficient of variation     statistics (ratio of standard deviation to expectation)
Correlation ρ or r   statistics (measure of linear dependence)
Courant–Friedrich–Levy number C or 𝜈   mathematics (numerical solutions of hyperbolic PDEs)[27]
Euler's number e   mathematics (base of the natural logarithm)
Feigenbaum constants  ,    
 
chaos theory (period doubling)[28]
Golden ratio     mathematics, aesthetics (long side length of self-similar rectangle)
Pi     mathematics (ratio of a circle's circumference to its diameter)
Radian measure rad   mathematics (measurement of planar angles, 1 radian = 180/π degrees)
Steradian measure sr measurement of solid angles

Geography, geology and geophysics

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Name Standard symbol Definition Field of application
Albedo     climatology, astronomy (reflectivity of surfaces or bodies)
Love numbers h, k, l geophysics (solidity of earth and other planets)
Porosity     geology, porous media (void fraction of the medium)
Rossby number Ro   geophysics (ratio of inertial to Coriolis force)

Sport

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Name Standard symbol Definition Field of application
Blondeau number     sport science, team sports[29]
Gain ratio bicycling (system of representing gearing; length traveled over length pedaled)[30]
Goal difference GD   Association football[31]
Runs Per Wicket Ratio RpW ratio   cricket[32]
Winning percentage Various, e.g.   or   Various sports

Other fields

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Name Standard symbol Definition Field of application
Capacity factor   energy
Cohesion number Coh   Chemical engineering, material science, mechanics (A scale to show the energy needed for detaching two solid particles)[33][34]
Cost of transport COT   energy efficiency, economics (ratio of energy input to kinetic motion)
Damping ratio     mechanics, electrical engineering (the level of damping in a system)
Darcy number Da   porous media (ratio of permeability to cross-sectional area)
Decibel dB acoustics, electronics, control theory (ratio of two intensities or powers of a wave)
Dukhin number Du   colloid science (ratio of electric surface conductivity to the electric bulk conductivity in heterogeneous systems)
Elasticity
(economics)
E   economics (response of demand or supply to price changes)
Fine-structure constant     quantum electrodynamics (QED) (coupling constant characterizing the strength of the electromagnetic interaction)
Gain electronics (signal output to signal input)
Havnes parameter     In Dusty plasma physics, ratio of the total charge   carried by the dust particles   to the charge carried by the ions  , with   the number density of particles
Helmholtz number     The most important parameter in duct acoustics. If   is the dimensional frequency, then   is the corresponding free field wavenumber and   is the corresponding dimensionless frequency [35]
Iribarren number Ir   wave mechanics (breaking surface gravity waves on a slope)
Load factor   energy
Lundquist number S   plasma physics (ratio of a resistive time to an Alfvén wave crossing time in a plasma)
Peel number NP   coating (adhesion of microstructures with substrate)[36]
Perveance K   charged particle transport (measure of the strength of space charge in a charged particle beam)
Pierce parameter     Traveling wave tube
Pixel px digital imaging (smallest addressable unit)
Beta (plasma physics)     Plasma (physics) and Fusion power. Ratio of plasma thermal pressure to magnetic pressure, controlling the level of turbulence in a magnetised plasma.
Poisson's ratio     elasticity (strain in transverse and longitudinal direction)
Power factor pf   electrical (real power to apparent power)
Power number Np   fluid mechanics, power consumption by rotary agitators; resistance force versus inertia force)
Prater number β   reaction engineering (ratio of heat evolution to heat conduction within a catalyst pellet)[37]
Q factor Q   physics, engineering (Damping ratio of oscillator or resonator; energy stored versus energy lost)
Relative density RD   hydrometers, material comparisons (ratio of density of a material to a reference material—usually water)
Relative permeability     magnetostatics (ratio of the permeability of a specific medium to free space)
Relative permittivity     electrostatics (ratio of capacitance of test capacitor with dielectric material versus vacuum)
Rouse number P or Z   sediment transport (ratio of the sediment fall velocity and the upwards velocity of grain)
Shields parameter   or     sediment transport (threshold of sediment movement due to fluid motion; dimensionless shear stress)
Specific gravity SG (same as Relative density)
Stefan number Ste   phase change, thermodynamics (ratio of sensible heat to latent heat)
Strain     materials science, elasticity (displacement between particles in the body relative to a reference length)

References

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  1. ^ a b c "Table of Dimensionless Numbers" (PDF). Retrieved 2009-11-05.
  2. ^ Becker, A.; Hüttinger, K. J. (1998). "Chemistry and kinetics of chemical vapor deposition of pyrocarbon—II pyrocarbon deposition from ethylene, acetylene and 1,3-butadiene in the low temperature regime". Carbon. 36 (3): 177. doi:10.1016/S0008-6223(97)00175-9.
  3. ^ Incropera, Frank P. (2007). Fundamentals of heat and mass transfer. John Wiley & Sons, Inc. p. 376. ISBN 9780470055540.
  4. ^ Popov, Konstantin I.; Djokić, Stojan S.; Grgur, Branimir N. (2002). Fundamental Aspects of Electrometallurgy. Boston, MA: Springer. pp. 101–102. ISBN 978-0-306-47564-1.
  5. ^ Kuneš, J. (2012). "Technology and Mechanical Engineering". Dimensionless Physical Quantities in Science and Engineering. pp. 353–390. doi:10.1016/B978-0-12-416013-2.00008-7. ISBN 978-0-12-416013-2.
  6. ^ Asakuma, Y. (2020). "A dimensionless number for microwave non-equilibrium local heating through surfactant desorption". Colloids and Surfaces A: Physicochemical and Engineering Aspects. 591: 124560. doi:10.1016/j.colsurfa.2020.124560.
  7. ^ Bagnold number Archived 2005-05-10 at the Wayback Machine
  8. ^ Bhattacharjee S.; Grosshandler W.L. (1988). "The formation of wall jet near a high temperature wall under microgravity environment". ASME MTD. 96: 711–6. Bibcode:1988nht.....1..711B.
  9. ^ Paoletti S.; Rispoli F.; Sciubba E. (1989). "Calculation of exergetic losses in compact heat exchanger passager". ASME AES. 10 (2): 21–9.
  10. ^ Bond number Archived 2012-03-05 at the Wayback Machine
  11. ^ "Home". OnePetro. 2015-05-04. Retrieved 2015-05-08.
  12. ^ Schetz, Joseph A. (1993). Boundary Layer Analysis. Englewood Cliffs, NJ: Prentice-Hall, Inc. pp. 132–134. ISBN 0-13-086885-X.
  13. ^ "Fanning friction factor". Archived from the original on 2013-12-20. Retrieved 2015-10-07.
  14. ^ Tan, R. B. H.; Sundar, R. (2001). "On the froth–spray transition at multiple orifices". Chemical Engineering Science. 56 (21–22): 6337. Bibcode:2001ChEnS..56.6337T. doi:10.1016/S0009-2509(01)00247-0.
  15. ^ Lockhart–Martinelli parameter
  16. ^ "Manning coefficient" (PDF). 10 June 2013. (109 KB)
  17. ^ Richardson number Archived 2015-03-02 at the Wayback Machine
  18. ^ Schmidt number Archived 2010-01-24 at the Wayback Machine
  19. ^ Sommerfeld number
  20. ^ Strouhal number, Engineering Toolbox
  21. ^ Straughan, B. (2001). "A sharp nonlinear stability threshold in rotating porous convection". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 457 (2005): 87–88. Bibcode:2001RSPSA.457...87S. doi:10.1098/rspa.2000.0657. S2CID 122753376.
  22. ^ Petritsch, G.; Mewes, D. (1999). "Experimental investigations of the flow patterns in the hot leg of a pressurized water reactor". Nuclear Engineering and Design. 188: 75–84. doi:10.1016/S0029-5493(99)00005-9.
  23. ^ Weissenberg number Archived 2006-11-01 at the Wayback Machine
  24. ^ Womersley number Archived 2009-03-25 at the Wayback Machine
  25. ^ Barbot, S. (2019). "Slow-slip, slow earthquakes, period-two cycles, full and partial ruptures, and deterministic chaos in a single asperity fault". Tectonophysics. 768: 228171. Bibcode:2019Tectp.76828171B. doi:10.1016/j.tecto.2019.228171.
  26. ^ Fresnel number Archived 2011-10-01 at the Wayback Machine
  27. ^ Courant–Friedrich–Levy number Archived 2008-06-05 at the Wayback Machine
  28. ^ Feigenbaum constants
  29. ^ Blondeau, J. (2021). "The influence of field size, goal size and number of players on the average number of goals scored per game in variants of football and hockey: the Pi-theorem applied to team sports". Journal of Quantitative Analysis in Sports. 17 (2): 145–154. doi:10.1515/jqas-2020-0009. S2CID 224929098.
  30. ^ Gain Ratio – Sheldon Brown
  31. ^ "goal difference". Cambridge Dictionary.
  32. ^ "World Test Championship Playing Conditions: What's different?" (PDF). International Cricket Council. Retrieved 11 August 2021.
  33. ^ Behjani, Mohammadreza Alizadeh; Rahmanian, Nejat; Ghani, Nur Fardina bt Abdul; Hassanpour, Ali (2017). "An investigation on process of seeded granulation in a continuous drum granulator using DEM" (PDF). Advanced Powder Technology. 28 (10): 2456–2464. doi:10.1016/j.apt.2017.02.011.
  34. ^ Alizadeh Behjani, Mohammadreza; Hassanpour, Ali; Ghadiri, Mojtaba; Bayly, Andrew (2017). "Numerical Analysis of the Effect of Particle Shape and Adhesion on the Segregation of Powder Mixtures". EPJ Web of Conferences. 140: 06024. Bibcode:2017EPJWC.14006024A. doi:10.1051/epjconf/201714006024. ISSN 2100-014X.
  35. ^ S.W. RIENSTRA, 2015, Fundamentals of Duct Acoustics, Von Karman Institute Lecture Notes
  36. ^ Van Spengen, W. M.; Puers, R.; De Wolf, I. (2003). "The prediction of stiction failures in MEMS". IEEE Transactions on Device and Materials Reliability. 3 (4): 167. doi:10.1109/TDMR.2003.820295.
  37. ^ Davis, Mark E.; Davis, Robert J. (2012). Fundamentals of Chemical Reaction Engineering. Dover. p. 215. ISBN 978-0-486-48855-4.

Bibliography

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