Dimensionless Numbers
Physical and chemical dimensionless quantities - Reynolds number, Euler, Nusselt, and Prandtl number - 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 = 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/(s2K))
f = frequency (1/s)
γ = surface tension (kg/s2)
x = mole fraction (1)
kd = mass transfer coefficient (m/s)
