Engineering ToolBox - Resources, Tools and Basic Information for Engineering and Design of Technical Applications!

# 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.

For full table - rotate the screen!

 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]

## Related Topics

• ### Basics

The SI-system, unit converters, physical constants, drawing scales and more.
• ### Miscellaneous

Engineering related topics like Beaufort Wind Scale, CE-marking, drawing standards and more.

## Related Documents

• ### Designation of Large Numbers

Designation of large number in US vs. other countries.
• ### Euler Number

Introduction to the Euler Number used in fluid mechanics.
• ### Froude Number

Introduction to the Froude Number.
• ### Mach Number

An introduction to the Mach Number.
• ### Numbers - Square, Cube, Square Root and Cubic Root Calculator

Calculate square, cube, square root and cubic root. Values tabulated for numbers ranging 1 to 100.
• ### Prandtl Number

A dimensionless number approximating the ratio of momentum diffusivity to thermal diffusivity.
• ### Reynolds Number

Introduction and definition of the dimensionless Reynolds Number - online calculators.
• ### Strouhal Number

Introduction to the Strouhal Number
• ### Symbols Used to Denote a Chemical Reactions and Process or Condition

Explanation of symbols used as subscripts or superscripts to tell more about the type of chemical reaction, process or condition.
• ### Thermodynamic Terms - Functions and Relations

Common thermodynamic terms and functions - potential energy, kinetic energy, thermal or internal energy, chemical energy, nuclear energy and more.
• ### Weber Number

The Weber Number may be useful when analyzing fluid flows where there is an interface between two different fluids.

## Search

Search is the most efficient way to navigate the Engineering ToolBox.

## Engineering ToolBox - SketchUp Extension - Online 3D modeling!

Add standard and customized parametric components - like flange beams, lumbers, piping, stairs and more - to your Sketchup model with the Engineering ToolBox - SketchUp Extension - enabled for use with older versions of the amazing SketchUp Make and the newer "up to date" SketchUp Pro . Add the Engineering ToolBox extension to your SketchUp Make/Pro from the Extension Warehouse !

We don't collect information from our users. More about

## Citation

• The Engineering ToolBox (2017). Dimensionless Numbers. [online] Available at: https://www.engineeringtoolbox.com/dimensionless-number-quantity-symbol-application-d_1982.html [Accessed Day Month Year].

Modify the access date according your visit.

12.8.9