Dimensionless Number

In dimensional analysis, a dimensionless number (or more precisely, a number with the dimensions of 1) is a quantity which describes a certain physical system and which is a pure number without any physical units; it does not change if one alters one's system of units of measurement, for example from English units to metric units. Such a number is typically defined as a product or ratio of quantities which do have units, in such a way that all units cancel. For example: "one out of every 10 apples I gather is rotten." The rotten-to-gathered ratio is (1 apple) / (10 apples) = 0.1, which is a dimensionless quantity. Dimensionless numbers are widely applied in the field of mechanical and chemical engineering. According to the Buckingham p-theorem of dimensional analysis, the functional dependence between a certain number (e.g.: n) of variables can be reduced by the number (e.g. k) of independent dimensions occurring in those variables to give a set of p = n - k independent, dimensionless numbers. For the purposes of the experimenter, different systems which share the same description by dimensionless numbers are equivalent.

Properties

  • A dimensionless number has no physical unit associated with it. However, it is sometimes helpful to use the same units in both the numerator and denominator, such as kg/kg, to show the quantity being measured.
  • A dimensionless number has the same value regardless of the measurement units used to calculate it. It has the same value whether it was calculated using the metric measurement system or the imperial measurement system.
  • However, a number may be dimensionless in one system of units (e.g., in a nonrationalized cgs system of units with the electric constant e0 = 1), and not dimensionless in another system of units (e.g., the rationalized SI system, with e0 = 8.85419×10-12 F/m).

An Example

The power-consumption of a stirrer with a particular geometry is a function of the density and the viscosity of the fluid to be stirred, the size of the stirrer given by its diameter, and the speed of the stirrer. Therefore, we have n = 5 variables representing our example. Those n = 5 variables are built up from k = 3 dimensions which are:

  • Length L [m]
  • Time T [s]
  • Mass M [kg]

According to the p-theorem, the n = 5 variables can be reduced by the k = 3 dimensions to form p = n - k = 5 - 3 = 2 independent dimensionless numbers which are in case of the stirrer

  • Reynolds number (This is the most important dimensionless number; it describes the fluid flow regime)
  • Power number (describes the stirrer and also involves the density of the fluid)

List of Dimensionless Numbers

There are infinitely many dimensionless numbers. Some of those that are used most often have been given names, as in the following list of examples (in alphabetical order, indicating their field of use):

  • Abbe number: Dispersion in optical materials
  • Archimedes number: Motion of fluids due to density differences
  • Biot number: Surface vs volume conductivity of solids
  • Bodenstein number: residence-time distribution
  • Capillary number: fluid flow influenced by surface tension
  • Damköhler numbers: reaction time scales vs transport phenomena
  • Deborah number: Rheology of viscoelastic fluids
  • Drag coefficient: Flow resistance
  • Eckert number: Convective heat transfer
  • Ekman number: Frictional (viscous) forces in geophysics
  • Euler number: Hydrodynamics (pressure forces vs. inertia forces)
  • Darcy Friction factor: Fluid flow
  • Froude number: Wave and surface behaviour
  • Grashof number: Free convection
  • Hagen number: Forced convection
  • Knudsen number: Continuum approximation in fluids
  • Laplace number: Free convection with immiscible fluids
  • Lift coefficient: amount of lift available from given airfoil at given angle of attack.
  • Mach number: Gas dynamics
  • Molecular mass
  • Nusselt number: Heat transfer with forced convection
  • Ohnesorge number: Atomization of liquids
  • Peclet number: Forced convection
  • Pressure coefficient: Coefficient of pressure experienced at a point on an airfoil
  • Poisson's ratio: Load in transverse and longitudinal direction
  • Power number: Power consumption by agitators
  • Prandtl number: Forced and free convection
  • Rayleigh number: Buoyancy and viscous forces in free convection
  • Reynolds number: Characterizing flow behaviour (laminar or turbulent)
  • Richardson number: whether buoyancy is important
  • Rockwell scale: Mechanical hardness
  • Rossby number: Inertial forces in geophysics
  • Sherwood number: Mass transfer with forced convection
  • Stokes number: Dynamics of particles
  • Strouhal number: Oscillatory flows
  • Weber number: Characterization of multiphase flow with strongly curved surfaces
  • Weissenberg number: Viscoelastic flows