Dimensionless number

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Dimensionless numbers in transport phenomena are a set of dimensionless quantities that have an important role in the calculations of momentum, mass and heat transfer.

Diffusive numbers in transport phenomena

Dimensionless numbers in transport phenomena
vs. Inertial Viscous Thermal Mass
Inertial vd Re Pe PeAB
Viscous Re−1 η, μ/ρ Pr Sc
Thermal Pe−1 Pr−1 α Le
Mass PeAB−1 Sc−1 Le−1 D

The classical numbers in transport phenomena of mass, momentum, and energy are principally analyzed by the ratio of effective diffusivities in each transport mechanism. The six dimensionless numbers give the relative strengths of the different phenomena of inertia, viscosity, conductive heat transport, and mass transport. (In the table, the number is the ratio of the left column quantity over top row quantity; e.g. Re = vd/η.) These same quantities may alternatively be expressed as ratios of characteristic time, length, or energy scales; these are less commonly used but can provide insight into particular applications.


All numbers are dimensionless quantities

Name Standard symbol Definition Word definition
Biot number Ratio of thermal resistance external to a boundary or surface to the internal thermal resistance
Brinkman number Ratio of heat transfer to viscous heating
Darcy friction factor Ratio of pressure drop to kinetic energy in pipe flow.
Drag coefficient Ratio of drag force to kinetic energy in external flow
Fanning friction factor Ratio of wall shear stress to kinetic energy in pipe flow.
Grashof number Ratio of bouyant to viscous forces in natural convection
Mach number Ratio of flow velocity in medium to speed of sound in medium
Nusselt number Ratio of convective to conductive heat transfer
Peclet number or Ratio of advective transport to molecular diffusion in mass transfer or the ratio of advective transport to thermal diffusion in heat transfer.
Prandtl number heat transfer (the ratio of viscous diffusion to thermal diffusion)
Rayleigh number heat transfer (buoyancy versus viscous forces in free convection)
Reynolds number fluid mechanics (ratio of fluid inertial and viscous forces)
Schmidt number mass transfer (viscous over molecular diffusion rate)
Sherwood number mass transfer (forced convection; ratio of convective to diffusive mass transport)