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 the units cancel out.
Contents
Examples
"out of every 10 apples I gather, 1 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. Angles are typically measured as the ratio of the length of an arc lying on a circle (with its center being the vertex of the angle) swept out by the angle, compared to some other length. The ratio (length divided by length) is dimensionless. When using the unit of "radians" the length that is compared is the length of the radius of the circle. When using the unit of "degrees" the length that is compared is 1/360 of the circumference of the circle.
Dimensionless quantities are widely used in the fields of mathematics, physics, engineering, and economics 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 = 1^{-6}). Dimensionless quantities can also carry dimensionless units like % (=0.01), ppt (=10^{-3}), ppm (=10^{-6}), ppb (=10^{-9}).
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]
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 SI system of units or the imperial system of units. This doesn't hold for all dimensionless quantities; it is guaranteed to hold only for proportions.
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.
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 a very 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 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 |
Colburn j factor | dimensionless heat transfer coefficient |
Courant-Friedrich-Levy number | numerical solutions of hyperbolic PDEs[7] |
Courtin number | torque on rotating fluids |
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 |
Decibel | ratio of two intensities of sound |
Drag coefficient | flow resistance |
e | mathematics |
Eckert number | convective heat transfer |
Ekman number | geophysics (frictional (viscous) forces) |
Elasticity (economics) | widely used to measure how demand or supply responds to price changes |
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] |
Fine structure constant | quantum electrodynamics (QED) |
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 |
Hatta number | adsorption enhancement due to chemical reaction |
Hagen number | forced convection |
Karlovitz number | turbulent combustion |
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 |
Magnetic Reynolds number | magnetohydrodynamics |
Manning roughness coefficient | open channel flow (flow driven by gravity) [12]PDF (109 KiB) |
Marangoni number | Marangoni flow due to thermal surface tension deviations |
Morton number | determination of bubble/drop shape |
Nusselt number | heat transfer with forced convection |
Ohnesorge number | atomization of liquids, Marangoni flow |
Péclet number | advection–diffusion problems |
Peel number | adhesion of microstructures with substrate [13] |
Pi | mathematics (ratio of a circle's circumference to its diameter) |
Poisson's ratio | elasticity (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 |
Pressure coefficient | pressure experienced at a point on an airfoil |
Radian | measurement of angles |
Rayleigh number | buoyancy and viscous forces in free convection |
Refractive index | electromagnetism, optics |
Reynolds number | flow behavior (inertia vs. viscosity) |
Relative density | hydrometers, material comparisons |
Richardson number | effect of buoyancy on flow stability [14] |
Rockwell scale | mechanical hardness |
Rossby number | inertial forces in geophysics |
Schmidt number | fluid dynamics (mass transfer and diffusion) [15] |
Sherwood number | mass transfer with forced convection |
Sommerfeld number | boundary lubrication [16] |
Stanton number | heat transfer in forced convection |
Stefan number | heat transfer during phase change |
Stokes number | particle dynamics |
Strain | materials science, elasticity |
Strouhal number | continuous and pulsating flow [17] |
Taylor number | rotating fluid flows |
van 't Hoff factor | quantitative analysis (K_{f} and K_{b}) |
Weaver flame speed number | laminar burning velocity relative to hydrogen gas [18] |
Weber number | multiphase flow with strongly curved surfaces |
Weissenberg number | viscoelastic flows [19] |
Womersley number | continuous and pulsating flows [20] |
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:
- , the fine structure constant and the electromagnetic coupling constant
- , 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
See also
- Similitude (model)
- Orders of magnitude (numbers)
- Dimensional analysis
- Normalization (statistics) and Standardized moment, the analogous concepts in statistics
External links
- Biographies of 16 scientists with dimensionless numbers of heat and mass transfer named after them
- How Many Fundamental Constants Are There? by John Baez
- NRL Plasma Formulary, Dimensionless Numbers of Fluid Mechanics, pp. p. 23, p. 24 and p. 25), J.D. Huba, Naval Research Laboratory (2007)
- Systematic Search for Expressions of Dimensionless Constants using the NIST database of Physical Constants Mike Sheppard, 2007
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