# Dimensionless numbers

## Definitions and symbols for physical and chemical dimensionless quantities, with areas of application of the different numbers. Reynolds, Euler, Nusselt, Prandtl..... and many more

The table shows the definitions of a lot of dimensionless quantities used in chemistry, fluid flow and physics engineering. Below the table, the symbols used in the formulas are explained and given with SI units.

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Name | Symbol | Formula | Areas of application |

Alfvén number | Al | Al = ν(ρ μ)^{½}/B | Study of magnetic fields |

Cowling number | Co | Co = B^{2 }/(μ ρ ν^{2}) | Study of magnetic fields |

Euler number | Eu | Eu = Δp /(ρ ν^{2}) | Characterization of energy losses in fluid flows |

Fourier number | Fo | Fo = a t / l^{2} | The ratio of diffusive or conductive heat transport rate to the heat storage rate |

Fourier number for mass transfer | Fo* | Fo* = D t / l^{2} | The ratio of diffusive mass transport rate to the mass storage rate |

Froude number | Fr | Fr = ν /(l g)^{½} | Determine the resistance of a partially submerged object moving through water |

Grashof number | Gr | Gr = l^{3} g α ΔT ρ^{2 }/ η^{2} | Study situations involving natural heat convection |

Grashof number for mass transfer | Gr* | Gr* = l^{3} g (∂p/∂x)_{T,p} (Δx p / η) | Predictions of mass flow patterns |

Hartmann number | Ha | Ha = B l (κ/η)^{1/2} | Describes the ratio of electromagnetic force to the viscous force |

Knudsen number | Kn | Kn = λ / l | Determine whether statistical mechanics or the continuum mechanics formulation of fluid dynamics should be used to model a situation |

Lewis number | Le | Le = a / D | Characterize fluid flows where there is simultaneous heat and mass transfer |

Mach number | Ma | Ma = ν / c | Determine the approximation with which a flow can be treated as an incompressible flow |

Nusselt number | Nu | Nu = h l / k | The ratio of convective to conductive heat transfer across (normal to) a boundary surface, predicts flow patterns. |

Nusselt number for mass transfer | Nu* | Nu* = k_{d} l_{ }/ D | Predicts mass flow patterns |

Peclet number | Pe | Pe = ν l / a | For transport phenomena in a continuum, the ratio of advective to diffusive heat transport rates, to decide the simplicity/complexity of computational models |

Peclet number for mass transfer | Pe* | Pe* = ν l / D | The ratio of advective to diffusive mass transport rates |

Prandtl number | Pr | Pr = η / (ρ a) | Determine the thermal conductivity of gases at high temperatures |

Rayleigh number | Ra | Ra = l^{3} g α ΔT ρ /(η a) | Predict if heat transfer appear as conduction or convection |

Reynolds number | Re | Re = p ν l / η | Predictions of fluid flow patterns |

Magnetic Reynolds number | Re_{m} | Re_{m }= ν μ κ l | Estimates of the relative effects of advection or induction of a magnetic field |

Schmidt number | Sc | Sc = η /(ρ D) | Characterization of fluid flows in which there are simultaneous momentum and mass diffusion convection processes |

Stanton number | St | St = h /(ρ ν c_{p}) | Characterization of heat transfer in forced convection flows, the ratio of heat transferred into a fluid to the thermal capacity of fluid |

Stanton number for mass transfer | St* | St* = k_{d }/ ν | To characterize mass transfer in forced convection flows |

Strouhal number | Sr | Sr = l f / ν | Describing oscillating flow mechanisms |

Weber number | We | We = ρ ν^{2 }l / γ | Analysing fluid flows where there is an interface between two different fluids |

where

ν = speed [m/s]

η = viscosity [kg/(m s)]

ρ = density, mass density, [kg/m^{3}]

m = mass [kg]

V = volume [m^{3}]

l = length [m]

a = thermal diffusivity [m^{2}/s]

t = time [s]

μ = permeability [kg m/(s^{2}A^{2})]

B = magnetic flux density [kg/(s^{2}A)]

Δp = pressure difference [kg/(m s^{2})]

g = acceleration of free fall [m/s^{2}]

α = cubic expansion coefficient [1/K]

ΔT = temperature difference

κ = electric conductivity [s^{3} A^{2}/(kg m^{3})]

λ = mean free path [m]

D = diffusion coefficient [m^{2}/s]

c = speed of sound [m/s]

h = coefficient of heat transfer [kg/(s^{3} K)]

k = thermal conductivity [kg m/(s^{3} K)]

c_{p} = specific heat apacity at constant pressure [kg m^{2}/(s^{2} K)]

f = frequency [1/s]

γ = surface tension [kg/s^{2}]

x = mole fraction [1]

k_{d }= mass transfer coefficient [m/s]