Dimensionless number

A dimensionless number is a quantity which describes a certain physical system and which is a pure number without any physical 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 π-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.

An example: stirrer[?]

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 π-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)
• Power number (Describes the stirrer and also involves the density of the fluid)

Listing of dimensionless numbers

There are literally thousands (to be precise: infinite) dimensionless numbers including those being used most often: (in alphabetical order, indicating their field of use)

• Archimedes number[?]: Motion of fluids due to density differences
• Biot number: Surface vs volume conductivity of solids
• Capillary number[?]: fluid flow influenced by surface tension
• Deborah number: Rheology of viscoelastic[?] fluids
• Drag coefficient: Flow resistance
• Euler number: Hydrodynamics (pressure forces vs. inertia forces)
• Friction factor[?]: Fluid Flow
• Froude number[?]: Wave and surface behaviour
• Grashof number: Free convection
• Laplace number[?]: Free convection with inmiscible fluids
• Mach number: Gasdynamics
• Nusselt number: Heat transfer with forced convection
• Ohnesorge number[?]: Atomization of liquids
• Peclet number: Forced convection
• Power number: Power consumption by agitators
• Prandtl number: Forced and Free convection
• Rayleigh number: Buoyancy and viscous forces in free convection
• Reynolds number: Characterizing the flow behaviour (laminar or turbulent)
• Sherwood number[?]: Mass transfer
• Stokes number[?]: Dynamics of particles
• Strouhal number: Oscillatory flows
• Weber number[?]: Characterization of mulitphase flow with strongly bended surfaces
• Weissenberg number: Viscoelastic[?] flows

External link

• http://ichmt.me.metu.edu.tr/dimensionless/ - Biographies of 16 scientists with dimensionless numbers of heat and mass transfer named after them