List of dimensionless quantities

This is a list of well-known dimensionless quantities illustrating their variety of forms and applications. The table does not include pure numbers, dimensionless ratios, or dimensionless physical constants; these topics are discussed in the article.

Name Standard symbol Definition Field of application
Abbe numberVoptics (dispersion in optical materials)
Activity coefficientchemistry (Proportion of "active" molecules or atoms)
Albedoclimatology, astronomy (reflectivity of surfaces or bodies)
Archimedes numberArfluid mechanics (motion of fluids due to density differences)
Arrhenius numberchemistry (ratio of activation energy to thermal energy)[1]
Atomic weightMchemistry (mass of atom over one atomic mass unit, u, where carbon-12 is exactly 12 u)
Atwood numberAfluid mechanics (onset of instabilities in fluid mixtures due to density differences)
Bagnold numberBafluid mechanics, geology (ratio of grain collision stresses to viscous fluid stresses in flow of a granular material such as grain and sand)[2]
Basic reproduction numbernumber of infections caused on average by an infectious individual over entire infectious periodepidemiology
Bejan number
(fluid mechanics)
Befluid mechanics (dimensionless pressure drop along a channel)[3]
Bejan number
(thermodynamics)
Bethermodynamics (ratio of heat transfer irreversibility to total irreversibility due to heat transfer and fluid friction)[4]
Bingham numberBmfluid mechanics, rheology (ratio of yield stress to viscous stress)[1]
Biot numberBiheat transfer (surface vs. volume conductivity of solids)
Blake numberBl or Bgeology, fluid mechanics, porous media (inertial over viscous forces in fluid flow through porous media)
Bodenstein numberBo or Bdchemistry (residence-time distribution; similar to the axial mass transfer Peclet number)[5]
Bond numberBogeology, fluid mechanics, porous media (buoyant versus capillary forces, similar to the Eötvös number) [6]
Brinkman numberBrheat transfer, fluid mechanics (conduction from a wall to a viscous fluid)
Brownell–Katz numberNBKfluid mechanics (combination of capillary number and Bond number) [7]
Capillary numberCaporous media, fluid mechanics (viscous forces versus surface tension)
Chandrasekhar numberQmagnetohydrodynamics (ratio of the Lorentz force to the viscosity in magnetic convection)
Colburn J factorsJM, JH, JDturbulence; heat, mass, and momentum transfer (dimensionless transfer coefficients)
Coefficient of kinetic frictionmechanics (friction of solid bodies in translational motion)
Coefficient of static frictionmechanics (friction of solid bodies at rest)
Coefficient of determinationstatistics (proportion of variance explained by a statistical model)
Coefficient of variationstatistics (ratio of standard deviation to expectation)
Cohesion number Coh Chemical engineering, material science, mechanics (A scale to show the energy needed for detaching two solid particles)[8][9]
Correlationρ or rstatistics (measure of linear dependence)
Cost of transportCOTenergy efficiency, economics (ratio of energy input to kinetic motion)
Courant–Friedrich–Levy numberC or 𝜈mathematics (numerical solutions of hyperbolic PDEs)[10]
Damkohler numberDachemistry (reaction time scales vs. residence time)
Damping ratiomechanics (the level of damping in a system)
Darcy friction factorCf or fDfluid mechanics (fraction of pressure losses due to friction in a pipe; four times the Fanning friction factor)
Darcy numberDaporous media (ratio of permeability to cross-sectional area)
Dean numberDturbulent flow (vortices in curved ducts)
Deborah numberDerheology (viscoelastic fluids)
DecibeldBacoustics, electronics, control theory (ratio of two intensities or powers of a wave)
Drag coefficientcdaeronautics, fluid dynamics (resistance to fluid motion)
Dukhin numberDucolloid science (ratio of electric surface conductivity to the electric bulk conductivity in heterogeneous systems)
Eckert numberEcconvective heat transfer (characterizes dissipation of energy; ratio of kinetic energy to enthalpy)
Ekman numberEkgeophysics (viscous versus Coriolis forces)
Elasticity
(economics)
Eeconomics (response of demand or supply to price changes)
Eötvös numberEofluid mechanics (shape of bubbles or drops)
Ericksen numberErfluid dynamics (liquid crystal flow behavior; viscous over elastic forces)
Euler numberEuhydrodynamics (stream pressure versus inertia forces)
Euler's numberemathematics (base of the natural logarithm)
Excess temperature coefficientheat transfer, fluid dynamics (change in internal energy versus kinetic energy)[11]
Fanning friction factorffluid mechanics (fraction of pressure losses due to friction in a pipe; 1/4th the Darcy friction factor)[12]
Feigenbaum constants,
chaos theory (period doubling)[13]
Fine-structure constantquantum electrodynamics (QED) (coupling constant characterizing the strength of the electromagnetic interaction)
f-numberfoptics, photography (ratio of focal length to diameter of aperture)
Föppl–von Kármán numbervirology, solid mechanics (thin-shell buckling)
Fourier numberFoheat transfer, mass transfer (ratio of diffusive rate versus storage rate)
Fresnel numberFoptics (slit diffraction)[14]
Froude numberFrfluid mechanics (wave and surface behaviour; ratio of a body's inertia to gravitational forces)
Gainelectronics (signal output to signal input)
Gain ratiobicycling (system of representing gearing; length traveled over length pedaled)[15]
Galilei numberGafluid mechanics (gravitational over viscous forces)
Golden ratiomathematics, aesthetics (long side length of self-similar rectangle)
Görtler numberGfluid dynamics (boundary layer flow along a concave wall)
Graetz numberGzheat transfer, fluid mechanics (laminar flow through a conduit; also used in mass transfer)
Grashof numberGrheat transfer, natural convection (ratio of the buoyancy to viscous force)
Hatta numberHachemical engineering (adsorption enhancement due to chemical reaction)
Hagen numberHgheat transfer (ratio of the buoyancy to viscous force in forced convection)
Havnes parameterIn 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 numberThe most important parameter in duct acoustics. If is the dimensional frequency, then is the corresponding free field wavenumber and is the corresponding dimensionless frequency [16]
Hydraulic gradientifluid mechanics, groundwater flow (pressure head over distance)
Iribarren numberIrwave mechanics (breaking surface gravity waves on a slope)
Jakob numberJachemistry (ratio of sensible to latent energy absorbed during liquid-vapor phase change)[17]
Karlovitz numberKaturbulent combustion (characteristic chemical time scale to Kolmogorov time scale)
Keulegan–Carpenter numberKCfluid dynamics (ratio of drag force to inertia for a bluff object in oscillatory fluid flow)
Knudsen numberKngas dynamics (ratio of the molecular mean free path length to a representative physical length scale)
Kt/VKt/Vmedicine (hemodialysis and peritoneal dialysis treatment; dimensionless time)
Kutateladze numberKufluid mechanics (counter-current two-phase flow)[18]
Laplace numberLafluid dynamics (free convection within immiscible fluids; ratio of surface tension to momentum-transport)
Lewis numberLeheat and mass transfer (ratio of thermal to mass diffusivity)
Lift coefficientCLaerodynamics (lift available from an airfoil at a given angle of attack)
Lockhart–Martinelli parametertwo-phase flow (flow of wet gases; liquid fraction)[19]
Love numbersh, k, lgeophysics (solidity of earth and other planets)
Lundquist numberSplasma physics (ratio of a resistive time to an Alfvén wave crossing time in a plasma)
Mach numberM or Magas dynamics (compressible flow; dimensionless velocity)
Magnetic Reynolds numberRmmagnetohydrodynamics (ratio of magnetic advection to magnetic diffusion)
Manning roughness coefficientnopen channel flow (flow driven by gravity)[20]
Marangoni numberMgfluid mechanics (Marangoni flow; thermal surface tension forces over viscous forces)
Markstein numberfluid dynamics, combustion (turbulent combustion flames)
Morton numberMofluid dynamics (determination of bubble/drop shape)
Nusselt numberNuheat transfer (forced convection; ratio of convective to conductive heat transfer)
Ohnesorge numberOhfluid dynamics (atomization of liquids, Marangoni flow)
Péclet numberPeheat transfer (advectiondiffusion problems; total momentum transfer to molecular heat transfer)
Peel numberNPcoating (adhesion of microstructures with substrate)[21]
PerveanceKcharged particle transport (measure of the strength of space charge in a charged particle beam)
pHchemistry (the measure of the acidity or basicity of an aqueous solution)
Pimathematics (ratio of a circle's circumference to its diameter)
Pierce parameterTraveling wave tube
Pixelpxdigital 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 ratioelasticity (strain in transverse and longitudinal direction)
Porositygeology, porous media (void fraction of the medium)
Power factorpf electrical (real power to apparent power)
Power numberNpelectronics (power consumption by agitators; resistance force versus inertia force)
Prandtl numberPrheat transfer (ratio of viscous diffusion rate over thermal diffusion rate)
Prater numberβreaction engineering (ratio of heat evolution to heat conduction within a catalyst pellet)[22]
Pressure coefficientCPaerodynamics, hydrodynamics (pressure experienced at a point on an airfoil; dimensionless pressure variable)
Q factorQphysics, engineering (damping of oscillator or resonator; energy stored versus energy lost)
Radian measureradmathematics (measurement of planar angles, 1 radian = 180/π degrees)
Rayleigh numberRaheat transfer (buoyancy versus viscous forces in free convection)
Refractive indexnelectromagnetism, optics (speed of light in a vacuum over speed of light in a material)
Relative densityRDhydrometers, material comparisons (ratio of density of a material to a reference material—usually water)
Relative permeabilitymagnetostatics (ratio of the permeability of a specific medium to free space)
Relative permittivityelectrostatics (ratio of capacitance of test capacitor with dielectric material versus vacuum)
Reynolds numberRefluid mechanics (ratio of fluid inertial and viscous forces)[1]
Richardson numberRifluid dynamics (effect of buoyancy on flow stability; ratio of potential over kinetic energy)[23]
Rockwell scalemechanical hardness (indentation hardness of a material)
Rolling resistance coefficientCrrvehicle dynamics (ratio of force needed for motion of a wheel over the normal force)
Roshko numberRofluid dynamics (oscillating flow, vortex shedding)
Rossby numberRogeophysics (ratio of inertial to Coriolis force)
Rouse numberP or Zsediment transport (ratio of the sediment fall velocity and the upwards velocity of grain)
Schmidt numberScmass transfer (viscous over molecular diffusion rate)[24]
Shape factorHboundary layer flow (ratio of displacement thickness to momentum thickness)
Sherwood numberShmass transfer (forced convection; ratio of convective to diffusive mass transport)
Shields parameter or sediment transport (threshold of sediment movement due to fluid motion; dimensionless shear stress)
Sommerfeld numberShydrodynamic lubrication (boundary lubrication)[25]
Specific gravitySG(same as Relative density)
Stanton numberStheat transfer and fluid dynamics (forced convection)
Stefan numberStephase change, thermodynamics (ratio of sensible heat to latent heat)
Stokes numberStk or Skparticles suspensions (ratio of characteristic time of particle to time of flow)
Strainmaterials science, elasticity (displacement between particles in the body relative to a reference length)
Strouhal numberSt or Srfluid dynamics (continuous and pulsating flow; nondimensional frequency)[26]
Stuart numberNmagnetohydrodynamics (ratio of electromagnetic to inertial forces)
Taylor numberTafluid dynamics (rotating fluid flows; inertial forces due to rotation of a fluid versus viscous forces)
TransmittanceToptics, spectroscopy (the ratio of the intensities of radiation exiting through and incident on a sample)
Ursell numberUwave mechanics (nonlinearity of surface gravity waves on a shallow fluid layer)
Vadasz numberVaporous media (governs the effects of porosity , the Prandtl number and the Darcy number on flow in a porous medium) [27]
van 't Hoff factoriquantitative analysis (Kf and Kb)
Wallis parameterj*multiphase flows (nondimensional superficial velocity)[28]
Wagner numberWaelectrochemistry (ratio of kinetic polarization resistance to solution ohmic resistance in an electrochemical cell)[29]
Weaver flame speed numberWeacombustion (laminar burning velocity relative to hydrogen gas)[30]
Weber numberWemultiphase flow (strongly curved surfaces; ratio of inertia to surface tension)
Weissenberg numberWiviscoelastic flows (shear rate times the relaxation time)[31]
Womersley numberbiofluid mechanics (continuous and pulsating flows; ratio of pulsatile flow frequency to viscous effects)[32]
Zel'dovich numberfluid dynamics, Combustion (Measure of activation energy)

References

  1. "Table of Dimensionless Numbers" (PDF). Retrieved 2009-11-05.
  2. Bagnold number Archived 2005-05-10 at the Wayback Machine
  3. 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.
  4. Paoletti S.; Rispoli F.; Sciubba E. (1989). "Calculation of exergetic losses in compact heat exchanger passager". ASME AES. 10 (2): 21–9.
  5. 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.
  6. Bond number Archived 2012-03-05 at the Wayback Machine
  7. "Home". OnePetro. 2015-05-04. Retrieved 2015-05-08.
  8. 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.
  9. 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.
  10. Courant–Friedrich–Levy number Archived 2008-06-05 at the Wayback Machine
  11. Schetz, Joseph A. (1993). Boundary Layer Analysis. Englewood Cliffs, NJ: Prentice-Hall, Inc. pp. 132–134. ISBN 0-13-086885-X.
  12. "Fanning friction factor". Archived from the original on 2013-12-20. Retrieved 2015-10-07.
  13. Feigenbaum constants
  14. Fresnel number Archived 2011-10-01 at the Wayback Machine
  15. Gain Ratio – Sheldon Brown
  16. S.W. RIENSTRA, 2015, Fundamentals of Duct Acoustics, Von Karman Institute Lecture Notes
  17. Incropera, Frank P. (2007). Fundamentals of heat and mass transfer. John Wiley & Sons, Inc. p. 376.
  18. Tan, R. B. H.; Sundar, R. (2001). "On the froth–spray transition at multiple orifices". Chemical Engineering Science. 56 (21–22): 6337. doi:10.1016/S0009-2509(01)00247-0.
  19. Lockhart–Martinelli parameter
  20. "Manning coefficient" (PDF). (109 KB)
  21. 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.
  22. Davis, Mark E.; Davis, Robert J. (2012). Fundamentals of Chemical Reaction Engineering. Dover. p. 215. ISBN 978-0-486-48855-4.
  23. Richardson number Archived 2015-03-02 at the Wayback Machine
  24. Schmidt number Archived 2010-01-24 at the Wayback Machine
  25. Sommerfeld number
  26. Strouhal number, Engineering Toolbox
  27. 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.
  28. 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.
  29. 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.
  30. 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.
  31. Weissenberg number Archived 2006-11-01 at the Wayback Machine
  32. Womersley number Archived 2009-03-25 at the Wayback Machine
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