# Dimensionless quantity

Dimensionless quantity

In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and in everyday life (such as in counting). Numerous well-known quantities, such as π, e, and φ, are dimensionless.

Dimensionless quantities are often defined as products or ratios of quantities that are not dimensionless, but whose dimensions cancel out when their powers are multiplied. This is the case, for instance, with the engineering strain, a measure of deformation. It is defined as change in length over initial length but, since these quantities both have dimensions L (length), the result is a dimensionless quantity.

## Properties

• Even though a dimensionless quantity has no physical dimension associated with it, it can still have dimensionless units (i.e. not unitless). To show the quantity being measured (for example mass fraction or mole fraction), it is sometimes helpful to use the same units in both the numerator and denominator (kg/kg or mol/mol). The quantity may also be given as a ratio of two different units that have the same dimension (for instance, light years over meters). This may be the case when calculating slopes in graphs, or when making unit conversions. Such notation does not indicate the presence of physical dimensions, and is purely a notational convention. Other common dimensionless units are  % (= 0.01),  ‰ (= 0.001), ppm (= 10−6), ppb (= 10−9), ppt (= 10−12) and angle units (degrees, radians, grad). Units of amount such as the dozen and the gross are also dimensionless.
• The ratio of two quantities with the same dimensions is dimensionless, and has the same value regardless of the units used to calculate them. For instance, if body A exerts a force of magnitude F on body B, and B exerts a force of magnitude f on A, then the ratio F/f will always be equal to 1, regardless of the actual units used to measure F and f. This is a fundamental property of dimensionless proportions and follows from the assumption that the laws of physics are independent of the system of units used in their expression. In this case, if the ratio F/f was not always equal to 1, but changed if we switched from SI to CGS, for instance, that would mean that Newton's Third Law's truth or falsity would depend on the system of units used, which would contradict this fundamental hypothesis. The assumption that the laws of physics are not contingent upon a specific unit system is also closely related to the Buckingham π theorem. A formulation of this theorem is that any physical law can be expressed as an identity (always true equation) involving only dimensionless combinations (ratios or products) of the variables linked by the law (e. g., pressure and volume are linked by Boyle's Law – they are inversely proportional). If the dimensionless combinations' values changed with the systems of units, then the equation would not be an identity, and Buckingham's theorem would not hold.

## Buckingham π theorem

Another consequence of the Buckingham π theorem of dimensional analysis is that the functional dependence between a certain number (say, n) of variables can be reduced by the number (say, k) of independent dimensions occurring in those variables to give a set of p = nk independent, dimensionless quantities. 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 given shape 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 (a dimensionless number describing the fluid flow regime)
• Power number (describing the stirrer and also involves the density of the fluid)

## Standards efforts

The International Committee for Weights and Measures contemplated defining the unit of 1 as the 'uno', but the idea was dropped.

## Examples

• Consider this example: Sarah says, "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. An angle is measured as the ratio of the length of a circle's arc subtended by an angle whose vertex is the centre of the circle to some other length. The ratio, length divided by length, is dimensionless. When using radians as the unit, the length that is compared is the length of the radius of the circle. When using degree as the units, the arc's length is compared to 1/360 of the circumference of the circle.
• In the case of the dimensionless quantity π, being the ratio of a circle's circumference to its diameter, the number would be constant regardless of what unit is used to measure a circle's circumference and diameter (eg. centimetres, miles, light-years, etc), as long as the same unit is used for both.

## List of dimensionless quantities

All numbers are dimensionless quantities. Certain dimensionless quantities of some importance are given below:

Name Standard symbol Definition Field of application
Abbe number V optics (dispersion in optical materials)
Activity coefficient γ chemistry (Proportion of "active" molecules or atoms)
Albedo α climatology, astronomy (reflectivity of surfaces or bodies)
Archimedes number Ar motion of fluids due to density differences
Arrhenius number α Ratio of activation energy to thermal energy
Atomic weight M chemistry
Bagnold number Ba flow of bulk solids such as grain and sand.
Blowdown circulation number BC deviation from isothermal flow in blowdown (rapid depressurization) of a pressure vessel
Bejan number
(thermodynamics)
Be the ratio of heat transfer irreversibility to total irreversibility due to heat transfer and fluid friction
Bejan number
(fluid mechanics)
Be dimensionless pressure drop along a channel
Bingham number Bm $Bm = \frac{ \tau_yL }{ \mu V }$ Ratio of yield stress to viscous stress
Bingham capillary number Bm.Ca $Bm.Ca = \frac{\tau_yL }{\gamma }$ Ratio of yield stress to capillary pressure
Biot number Bi surface vs. volume conductivity of solids
Blake number Bl or B relative importance of inertia compared to viscous forces in fluid flow through porous media
Bodenstein number Bo $Bo = Re\cdot Sc = vL/\mathcal{D}$ residence-time distribution
Bond number Bo capillary action driven by buoyancy 
Brinkman number Br heat transfer by conduction from the wall to a viscous fluid
Brownell–Katz number combination of capillary number and Bond number
Capillary number Ca fluid flow influenced by surface tension
Coefficient of static friction μs friction of solid bodies at rest
Coefficient of kinetic friction μk friction of solid bodies in translational motion
Colburn j factor dimensionless heat transfer coefficient
Courant–Friedrich–Levy number ν numerical solutions of hyperbolic PDEs 
Damkohler number Da Da = kτ reaction time scales vs. resonance time
Damping ratio ζ $\zeta = \frac{c}{2 \sqrt{km}}$ the level of damping in a system
Darcy friction factor Cf or f fluid flow
Dean number D vortices in curved ducts
Deborah number De rheology of viscoelastic fluids
Decibel dB ratio of two intensities, often sound
Drag coefficient Cd flow resistance
Dukhin number Du ratio of electric surface conductivity to the electric bulk conductivity in heterogeneous systems
Euler's number e mathematics
Eckert number Ec convective heat transfer
Ekman number Ek geophysics (frictional (viscous) forces)
Elasticity (economics) E widely used to measure how demand or supply responds to price changes
Eötvös number Eo determination of bubble/drop shape
Ericksen number Er liquid crystal flow behavior
Euler number Eu hydrodynamics (pressure forces vs. inertia forces)
Fanning friction factor f fluid flow in pipes 
Feigenbaum constants α,δ chaos theory (period doubling) 
Fine structure constant α $\alpha=\frac{e^2}{2\varepsilon_0 hc}$ quantum electrodynamics (QED)
f-number f optics, photography
Foppl–von Karman number thin-shell buckling
Fourier number Fo heat transfer
Fresnel number F slit diffraction 
Froude number Fr $Fr = \frac{V}{\sqrt{g\ell}} \Rightarrow \frac{(intertia force)}{(gravitational force)}$ wave and surface behaviour
Gain electronics (signal output to signal input)
Galilei number Ga gravity-driven viscous flow
Golden ratio φ mathematics and aesthetics
Graetz number Gz heat flow
Grashof number Gr free convection
Gravitational coupling constant αG $\alpha_G=\frac{Gm_e^2}{\hbar c}$ Gravitation
Hatta number Ha adsorption enhancement due to chemical reaction
Hagen number Hg forced convection
Jakob Number Ja $Ja = \frac{c_p (T_s - T_{sat}) }{h_{fg} }$ Ratio of sensible to latent energy absorbed during liquid-vapor phase change
Karlovitz number turbulent combustion turbulent combustion
Keulegan–Carpenter number KC ratio of drag force to inertia for a bluff object in oscillatory fluid flow
Knudsen number Kn ratio of the molecular mean free path length to a representative physical length scale
Kt/V medicine
Kutateladze number K counter-current two-phase flow
Laplace number La free convection within immiscible fluids
Lewis number Le ratio of mass diffusivity and thermal diffusivity
Lift coefficient CL lift available from an airfoil at a given angle of attack
Lockhart–Martinelli parameter χ flow of wet gases 
Love number measuring the solidity of the earth
Lundquist number S ratio of a resistive time to an Alfvén wave crossing time in a plasma
Mach number M Ratio of current speed to the speed of sound, i.e. Mach 1 is the speed of sound, Mach 0.5 is half the speed of sound, Mach 2 is twice the speed of sound. gas dynamics
Magnetic Reynolds number Rm magnetohydrodynamics
Manning roughness coefficient n open channel flow (flow driven by gravity) 
Marangoni number Mg Marangoni flow due to thermal surface tension deviations
Morton number Mo determination of bubble/drop shape
Mpemba number KM thermal conduction and diffusion in freezing of a solution
Nusselt number Nu $Nu =\frac{hd}{k}$ heat transfer with forced convection
Ohnesorge number Oh atomization of liquids, Marangoni flow
Péclet number Pe $Pe = \frac{du\rho c_p}{k} = (Re)(Pr)$ advectiondiffusion problems; relates total momentun transfer to molecular heat transfer.
Peel number adhesion of microstructures with substrate 
Perveance K measure of the strength of space charge in a charged particle beam
Pi π mathematics (ratio of a circle's circumference to its diameter)
Poisson's ratio ν elasticity (load in transverse and longitudinal direction)
Porosity ϕ geology
Power factor electronics (real power to apparent power)
Power number Np power consumption by agitators
Prandtl number Pr $\Pr = \frac{\nu}{\alpha} = \frac{c_p \mu}{k}$ convection heat transfer (thickness of thermal and momentum boundary layers)
Pressure coefficient CP pressure experienced at a point on an airfoil
Q factor Q describes how under-damped an oscillator or resonator is
Rayleigh number Ra buoyancy and viscous forces in free convection
Refractive index n electromagnetism, optics
Reynolds number Re $Re = \frac{vL\rho}{\mu}$ Ratio of fluid inertial and viscous forces
Relative density RD hydrometers, material comparisons
Richardson number Ri effect of buoyancy on flow stability 
Rockwell scale mechanical hardness
Rolling resistance coefficient Crr $C_{rr} = \frac{N_f}{F}$ vehicle dynamics
Rossby number Ro inertial forces in geophysics
Rouse number Z or P sediment transport
Schmidt number Sc fluid dynamics (mass transfer and diffusion) 
Shape factor H ratio of displacement thickness to momentum thickness in boundary layer flow
Sherwood number Sh mass transfer with forced convection
Shields parameter τ or θ threshold of sediment movement due to fluid motion
Sommerfeld number boundary lubrication 
Stanton number St heat transfer in forced convection
Stefan number Ste heat transfer during phase change
Stokes number Stk or Sk particle dynamics in a fluid stream
Strain $\epsilon$ materials science, elasticity
Strouhal number St or Sr nondimensional frequency, continuous and pulsating flow 
Taylor number Ta rotating fluid flows
Ursell number U nonlinearity of surface gravity waves on a shallow fluid layer
Vadasz number Va $Va = \frac{\phi Pr}{Da}$ governs the effects of porosity ϕ, the Prandtl number and the Darcy number on flow in a porous medium
van 't Hoff factor i quantitative analysis (Kf and Kb)
Wallis parameter J* nondimensional superficial velocity in multiphase flows
Weaver flame speed number laminar burning velocity relative to hydrogen gas 
Weber number We multiphase flow with strongly curved surfaces
Weissenberg number Wi viscoelastic flows 
Womersley number α continuous and pulsating flows 

## Dimensionless physical constants

Certain fundamental physical constants, such as the speed of light in a vacuum, the universal gravitational constant, and the constants of Planck and Boltzmann, 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 natural. However, not all physical constants can be eliminated in any system of units; the values of the remaining ones must be determined experimentally. Resulting constants include:

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