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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.

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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 = 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

where

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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