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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

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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.

Name Symbol Formula Areas of application
Alfvén number Al Al = ν(ρ μ)½/B Study of magnetic fields
Cowling number Co Co = B2 /(μ ρ ν2) Study of magnetic fields
Euler number Eu Eu = Δp /(ρ ν2) Characterization of energy losses in fluid flows
Fourier number Fo Fo = a t / l2 The ratio of diffusive or conductive heat transport rate to the heat storage rate
Fourier number for mass transfer Fo* Fo* = D t / l2 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 = l3 g α ΔT ρ2 / η2 Study situations involving natural heat convection
Grashof number for mass transfer Gr* Gr* = l3 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* = kd 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 = l3 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 Rem Rem = 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 cp) 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* = kd / 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 ν2 l / γ Analysing fluid flows where there is an interface between two different fluids


ν = speed  [m/s]
η = viscosity  [kg/(m s)]
ρ = density, mass density, [kg/m3]
m = mass [kg]
V = volume [m3]
l = length [m]
a = thermal diffusivity  [m2/s]
t = time [s]
μ = permeability [kg m/(s2A2)]
B = magnetic flux density [kg/(s2A)]
Δp = pressure difference  [kg/(m s2)]
g = acceleration of free fall [m/s2]
α = cubic expansion coefficient [1/K]
ΔT = temperature difference
κ = electric conductivity [s3 A2/(kg m3)]
λ = mean free path [m]
D = diffusion coefficient [m2/s]
c = speed of sound [m/s]
h = coefficient of heat transfer [kg/(s3 K)]
k = thermal conductivity [kg m/(s3 K)]
cp = specific heat apacity at constant pressure  [kg m2/(s2 K)]
f = frequency [1/s]
γ = surface tension [kg/s2]
x = mole fraction [1]
kd = mass transfer coefficient  [m/s]

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