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Dimensionless number

In the physical sciences, 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 = nk independent, dimensionless numbers. For the purposes of the experimenter, different systems which share the same description by dimensionless numbers are equivalent. When plotting graphs, the quantities plotted are also dimensionless - the notation x (/s), for example, serves to divide the data (time in seconds) by the unit vector t, and produces a dimensionless value for plotting.


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 π-theorem, the n=5 variables can be reduced by the k=3 dimensions to form p = nk = 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):

Dimensionless physical constants

The system of natural units chooses its base units in such a way as to make several physical constants such as the speed of light into simple dimensionless constants by definition. However, other dimensionless physical constants cannot be eliminated, and have to be discovered experimentally. These are often called fundamental physical constants.

These include:

  • the fine structure constant
  • the electromagnetic coupling constant
  • the strong coupling constant
  • the gravitational fine structure constant

See also

External link

Last updated: 10-24-2004 05:10:45