Dimensionless quantity
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In dimensional analysis, a dimensionless quantity (or more precisely, a quantity with the dimensions of 1) is a quantity without any physical units and thus a pure number. 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.
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[edit] Examples
"one out of every 10 apples I gather is rotten." -- the rotten-to-gathered ratio is (1 apple) / (10 apples) = 0.1 = 10%, which is a dimensionless quantity. Another more typical example in physics and engineering is the measure of plane angles with the unit of "radian". An angle measured this way is the length of arc lying on a circle (with center being the vertex of the angle) swept out by the angle to the length of the radius of the circle. The units of the ratio is length divided by length which is dimensionless.
Dimensionless quantities are widely used in the fields of mathematics, physics, and engineering but also in everyday life. Whenever one measures any physical quantity, they are measuring that physical quantity against a like dimensioned standard. Whenever one commonly measures a length with a ruler or tape measure, they are counting tick marks on the standard of length they are using, which is a dimensionless number. When they attach that dimensionless number (the number of tick marks) to the units that the standard represents, they conceptually are referring to a dimensionful quantity. A quantity Q is defined as the product of that dimensionless number n (the number of tick marks) and the unit U (the standard):
But, ultimately, people always work with dimensionless numbers in reading measuring instruments and manipulating (changing or calculating with) even dimensionful quantities.
In case of dimensionless quantities the unit U is a quotient of like dimensioned quantities that can be reduced to a number (kg/kg = 1, μg/g = 1e-6). Dimensionless quantities can also carry dimensionless units like % (=0.01), ppm (=1e-6), ppb (=1e-9), ppt (=1e-12).
The CIPM Consultative Committee for Units toyed with the idea of defining the unit of 1 as the 'uno', but the idea was dropped. [1] [2] [3] [4]
[edit] Properties
- A dimensionless quantity 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 proportion 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. This doesn't hold for all dimensionless quantities; it is guaranteed to hold only for proportions.
- However, a physical quantity may be dimensionless in one system of units and not dimensionless in another system of units. For example, in the nonrationalized cgs system of units, the unit of electric charge (the statcoulomb) is defined in such a way so that the permittivity of free space ε0 = 1/(4π) whereas in the rationalized SI system, it is ε0 = 8.85419×10-12 F/m. In systems of natural units (e.g. Planck units or atomic units), the physical units are defined in such a way that several dimensioned physical constants are made dimensionless and set to 1 (thus removing these scaling factors from equations). While this is convenient in some contexts, abolishing of all or most units and dimensions often makes practical physical calculations more error prone.
[edit] Buckingham π-theorem
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 quantity. For the purposes of the experimenter, different systems which share the same description by dimensionless quantity are equivalent.
[edit] 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 = 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)
[edit] List of dimensionless quantities
There are infinitely many dimensionless quantities and they are often called numbers. Some of those that are used most often have been given names, as in the following list of examples (alphabetical order):
| Name | Field of application |
|---|---|
| Abbe number | optics (dispersion in optical materials) |
| Albedo | climatology, astronomy (reflectivity of surfaces or bodies) |
| Archimedes number | motion of fluids due to density differences |
| Bagnold number | flow of grain, sand, etc. [5] |
| Biot number | surface vs. volume conductivity of solids |
| Bodenstein number | residence-time distribution |
| Bond number | capillary action driven by buoyancy [6] |
| Brinkman number | heat transfer by conduction from the wall to a viscous fluid |
| Brownell Katz number | combination of capillary number and Bond number |
| Capillary number | fluid flow influenced by surface tension |
| Coefficient of static friction | friction of solid bodies at rest |
| Coefficient of kinetic friction | friction of solid bodies in translational motion |
| Courant-Friedrich-Levy number | non-hydrostatic dynamics [7] |
| Damköhler numbers | reaction time scales vs. transport phenomena |
| Darcy friction factor | fluid flow |
| Dean number | vortices in curved ducts |
| Deborah number | rheology of viscoelastic fluids |
| Drag coefficient | flow resistance |
| Eckert number | convective heat transfer |
| Ekman number | geophysics (frictional (viscous) forces) |
| Eötvös number | determination of bubble/drop shape |
| Euler number | hydrodynamics (pressure forces vs. inertia forces) |
| Fanning friction factor | fluid flow in pipes [8] |
| Feigenbaum constants | chaos theory (period doubling) [9] |
| Foppl–von Karman number | thin-shell buckling |
| Fourier number | heat transfer |
| Fresnel number | slit diffraction [10] |
| Froude number | wave and surface behaviour |
| Gain | electronics (signal output to signal input) |
| Galilei number | gravity-driven viscous flow |
| Graetz number | heat flow |
| Grashof number | free convection |
| Hagen number | forced convection |
| Knudsen number | continuum approximation in fluids |
| Kt/V | medicine |
| Laplace number | free convection within immiscible fluids |
| Lewis number | ratio of mass diffusivity and thermal diffusivity |
| Lockhart-Martinelli parameter | flow of wet gases [11] |
| Lift coefficient | lift available from an airfoil at a given angle of attack |
| Mach number | gas dynamics |
| Manning roughness coefficient | open channel flow (flow driven by gravity) [12]PDF (109 KiB) |
| Marangoni number | flow due to thermal surface tension deviations |
| Nusselt number | heat transfer with forced convection |
| Ohnesorge number | atomization of liquids |
| Péclet number | inertial forces vs. Brownian forces |
| Peel number | adhesion of microstructures with substrate [13] |
| Pressure coefficient | pressure experienced at a point on an airfoil |
| Poisson's ratio | load in transverse and longitudinal direction |
| Power factor | electronics (real power to apparent power) |
| Power number | power consumption by agitators |
| Prandtl number | forced and free convection |
| Radian | measurement of angles |
| Rayleigh number | buoyancy and viscous forces in free convection |
| Reynolds number | flow behaviour |
| magnetohydrodynamics [14] | |
| Richardson number | effect of buoyancy on flow stability [15] |
| Rockwell scale | mechanical hardness |
| Rossby number | inertial forces in geophysics |
| Schmidt number | fluid dynamics (mass transfer and diffusion) [16] |
| Sherwood number | mass transfer with forced convection |
| Sommerfeld number | boundary lubrication [17] |
| Stanton number | heat transfer in forced convection |
| Stefan number | heat transfer during phase change |
| Stokes number | particle dynamics |
| Strouhal number | continuous and pulsating flow [18] |
| van 't Hoff factor | quantitative analysis (Kf and Kb) |
| Weaver flame speed number | laminar burning velocity relative to hydrogen gas [19] |
| Weber number | multiphase flow with strongly curved surfaces |
| Weissenberg number | viscoelastic flows [20] |
| Womersley number | continuous and pulsating flows [21] |
[edit] Dimensionless physical constants
Certain physical constants, such as the speed of light in a vacuum, are normalized to 1 if the units for time, length, mass, charge, and temperature are chosen appropriately. The resulting system of units is known as Planck units. However, a handful of dimensionless physical constants cannot be eliminated in any system of units; their values must be determined experimentally. The resulting fundamental physical constants include:
- α, the fine structure constant and the electromagnetic coupling constant
- β, the ratio of the rest mass of the proton to that of the electron
- more generally, the masses of all fundamental particles relative to that of the electron
- the strong coupling constant
- the gravitational coupling constant
