# 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.[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. By contrast, non-dimensionless quantities are measured in units of length, area, time, etc.

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. 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 number 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 is always 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, 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. This assumption that the laws of physics are not contingent upon a specific unit system is the basis for the Buckingham π theorem. A statement of this theorem is that any physical law can be expressed as an identity 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 that 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:

• 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.[2][3][4]

## 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.
• 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—i.e., 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 (e.g., 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 $V = \frac{ n_d - 1 }{ n_F - n_C }$ optics (dispersion in optical materials)
Activity coefficient γ $\gamma= \frac {{a}}{{x}}$ chemistry (Proportion of "active" molecules or atoms)
Albedo $\alpha$ ${\alpha}= (1-D) \bar \alpha(\theta_i) + D \bar{ \bar \alpha}$ climatology, astronomy (reflectivity of surfaces or bodies)
Archimedes number Ar $Ar = \frac{g L^3 \rho_\ell (\rho - \rho_\ell)}{\mu^2}$ motion of fluids due to density differences
Arrhenius number $\alpha$ ratio of activation energy to thermal energy[5]
Atomic weight M chemistry
Bagnold number Ba $Ba = \frac{\rho d^2 \lambda^{1/2} \gamma}{\mu}$ flow of bulk solids such as grain and sand.[6]
Blowdown circulation number BC deviation from isothermal flow in blowdown (rapid depressurization) of a pressure vessel[7]
Bejan number
(thermodynamics)
Be $Be = \frac {\dot S'_{gen, \Delta T}} {\dot S'_{gen, \Delta T}+ \dot S'_{gen, \Delta p}}$ the ratio of heat transfer irreversibility to total irreversibility due to heat transfer and fluid friction[8]
Bejan number
(fluid mechanics)
Be $Be = \frac{\Delta P . L^2} {\mu \alpha}$ dimensionless pressure drop along a channel[9]
Bingham number Bm $Bm = \frac{ \tau_yL }{ \mu V }$ ratio of yield stress to viscous stress[5]
Bingham capillary number Bm.Ca $Bm.Ca = \frac{\tau_yL }{\gamma }$ ratio of yield stress to capillary pressure[10]
Biot number Bi $Bi = \frac{h L_C}{\ k_b}$ surface vs. volume conductivity of solids
Blake number Bl or B $B = \frac{u \rho}{\mu ( 1-\epsilon) D}$ 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 $Bo = \frac{\rho a L^2}{\gamma}$ capillary action driven by buoyancy[11]
Brinkman number Br $Br = \frac {\mu U^2}{\kappa(T_w-T_0)}$ 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
Chandrasekhar number $\ Q$ ${Q}\ =\ \frac{{B_0}^2 d^2}{\mu_0 \rho \nu \lambda}$ magnetic convection to represent ratio of the Lorentz force to the viscosity
Coefficient of static friction $\mu_s$ friction of solid bodies at rest
Coefficient of kinetic friction $\mu_k$ friction of solid bodies in translational motion
Colburn j factor dimensionless heat transfer coefficient
Courant–Friedrich–Levy number $\nu$ numerical solutions of hyperbolic PDEs[12]
Damkohler number Da $Da = k \tau$ reaction time scales vs. residence time
Damping ratio $\zeta$ $\zeta = \frac{c}{2 \sqrt{km}}$ the level of damping in a system
Darcy friction factor $C_f$ or $f$ fluid flow
Dean number D vortices in curved ducts
Deborah number De rheology of viscoelastic fluids
Decibel dB ratio of two intensities or powers, often sound
Drag coefficient $C_d$ 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
Entropy $\sigma$ as used in physics, log of number of states[13]
Eötvös number Eo determination of bubble/drop shape
Ericksen number Er liquid crystal flow behavior
Euler number Eu hydrodynamics (pressure forces vs. inertia forces)
Excess temperature coefficient Θr $\Theta_r = \frac{T-T_e}{U_e^2/(2c_p)}$ thermal and fluid dynamics[14]
Fanning friction factor f fluid flow in pipes[15]
Feigenbaum constants $\alpha, \delta$ chaos theory (period doubling)[16]
Fine structure constant $\alpha$ $\alpha = \frac{e^2}{2\varepsilon_0 hc}$ quantum electrodynamics (QED)
f-number $f$ $\frac {{\ell}}{{D}}$ ratio of focal length to diameter of aperture, optics, photography
Foppl–von Karman number thin-shell buckling
Fourier number Fo heat transfer
Fresnel number F slit diffraction[17]
Froude number Fr $Fr = \frac{v}{\sqrt{g\ell}}$ wave and surface behaviour
Gain electronics (signal output to signal input)
Gain ratio system of representing bicycle gearing[18]
Galilei number Ga gravity-driven viscous flow
Golden ratio $\varphi$ mathematics and aesthetics
Graetz number Gz heat flow
Grashof number Gr free convection
Gravitational coupling constant $\alpha_G$ $\alpha_G=\frac{Gm_e^2}{\hbar c}$ gravitation
Hatta number Ha adsorption enhancement due to chemical reaction
Hagen number Hg forced convection
Iribarren number Ir breaking surface gravity waves on a slope
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[19]
Karlovitz number turbulent combustion
Keulegan–Carpenter number $K_C$ 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 $C_L$ lift available from an airfoil at a given angle of attack
Lockhart–Martinelli parameter $\chi$ flow of wet gases[20]
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 or Ma $\ M = \frac {{v}}{{v_{sound}}}$ gas dynamics
Magnetic Reynolds number $R_m$ magnetohydrodynamics
Manning roughness coefficient n open channel flow (flow driven by gravity)[21]
Marangoni number Mg Marangoni flow due to thermal surface tension deviations
Morton number Mo determination of bubble/drop shape
Mpemba number $K_M$ thermal conduction and diffusion in freezing of a solution[22]
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 momentum transfer to molecular heat transfer.
Peel number adhesion of microstructures with substrate[23]
Perveance K measure of the strength of space charge in a charged particle beam
Pi $\pi$ mathematics (ratio of a circle's circumference to its diameter)
Poisson's ratio $\nu$ elasticity (load in transverse and longitudinal direction)
Porosity $\phi$ geology
Power factor electronics (real power to apparent power)
Power number $N_p$ power consumption by agitators
Prandtl number Pr $Pr = \frac{\nu}{\alpha} = \frac{c_p \mu}{k}$ ratio of viscous diffusion rate over thermal diffusion rate
Prater number ratio of heat evolution to heat conduction within a catalyst pellet
Pressure coefficient $C_P$ pressure experienced at a point on an airfoil
Q factor $Q$ describes how under-damped an oscillator or resonator is
Radian measure rad $arc~length/radius$ measurement of planar angles, 1 radian = 180/π degrees
Rayleigh number Ra buoyancy and viscous forces in free convection
Refractive index n electromagnetism, optics
Relative density RD $RD = \frac{\rho_\mathrm{substance}}{\rho_\mathrm{reference}}$ hydrometers, material comparisons
Relative permeability $\mu_{r}$ electrodynamics (e.g., magnetostatics)
Relative permittivity $\varepsilon_{r}$ electrodynamics (e.g., electrostatics)
Reynolds number Re $Re = \frac{vL\rho}{\mu}$ ratio of fluid inertial and viscous forces[5]
Richardson number Ri effect of buoyancy on flow stability[24]
Rockwell scale mechanical hardness
Rolling resistance coefficient Crr $C_{rr} = \frac{N_f}{F}$ vehicle dynamics
Rossby number $R_o$ inertial forces in geophysics
Rouse number Z or P sediment transport
Schmidt number Sc fluid dynamics (mass transfer and diffusion)[25]
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[26]
Specific gravity same as Relative density
Stanton number St heat transfer in forced convection
Stefan number Ste $Ste = \frac{C_p\Delta T}{L}$ heat transfer during phase change
Stokes number Stk or $S_k$ $Stk = \frac{\tau\,U_o}{d_c}$ particle dynamics in a fluid stream
Strain $\epsilon$ $\epsilon = \cfrac{\partial{F}}{\partial{X}} - 1$ materials science, elasticity
Strouhal number St or Sr $St = {\omega L\over v}$ nondimensional frequency, continuous, and pulsating flow[27]
Taylor number Ta $Ta = \frac{4\Omega^2 R^4}{\nu^2}$ rotating fluid flows
Ursell number U $U = \frac{H\, \lambda^2}{h^3}$ nonlinearity of surface gravity waves on a shallow fluid layer
Vadasz number Va $Va = \frac{\phi Pr}{Da}$ governs the effects of porosity $\phi$, the Prandtl number and the Darcy number on flow in a porous medium
van 't Hoff factor i $i = 1 + \alpha (n - 1)$ quantitative analysis (Kf and Kb)
Wallis parameter J* $\alpha = R \left( \frac{\omega \rho}{\mu} \right)^\frac{1}{2}$ nondimensional superficial velocity in multiphase flows
Weaver flame speed number laminar burning velocity relative to hydrogen gas[28]
Weber number We $We = \frac{\rho v^2 l}{\sigma}$ multiphase flow with strongly curved surfaces
Weissenberg number Wi $Wi = \dot{\gamma} \lambda$ viscoelastic flows[29]
Womersley number $\alpha$ $\alpha = R \left( \frac{\omega \rho}{\mu} \right)^\frac{1}{2}$ continuous and pulsating flows[30]

## 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:

## References

1. ^ "1.8 (1.6) quantity of dimension one dimensionless quantity". International vocabulary of metrology — Basic and general concepts and associated terms (VIM). ISO. 2008. Retrieved 2011-03-22.
2. ^ "BIPM Consultative Committee for Units (CCU), 15th Meeting" (PDF). 17–18 April 2003. Retrieved 2010-01-22.
3. ^ "BIPM Consultative Committee for Units (CCU), 16th Meeting" (PDF). Retrieved 2010-01-22.
4. ^ Dybkaer, René (2004). "An ontology on property for physical, chemical, and biological systems". APMIS Suppl. (117): 1–210. PMID 15588029.
5. ^ a b c "Table of Dimensionless Numbers" (PDF). Retrieved 2009-11-05.
6. ^ Bagnold number
7. ^ Katz J. I. (2009). "Circulation in blowdown flows". J. Pressure Vessel Technology 131 (3): 034501. doi:10.1115/1.3110038.
8. ^ Paoletti S., Rispoli F., Sciubba E. (1989). "Calculation of exergetic losses in compact heat exchanger passager". ASME AES 10 (2): 21–9.
9. ^ Bhattacharjee S., Grosshandler W.L. (1988). "The formation of wall jet near a high temperature wall under microgravity environment". ASME MTD 96: 711–6.
10. ^ German G., Bertola V. (2010). "The spreading behaviour of capillary driven yield-stress drops". Colloid Surface A 366: 18–26. doi:10.1016/j.colsurfa.2010.05.019.
11. ^ Bond number
12. ^ Courant–Friedrich–Levy number
13. ^ Kittel, Charles and Herbert Kroemer (1980). Thermal Physics. New York: W. H. Freeman. p. 1. ISBN 0-7167-1088-9.
14. ^ Schetz, Joseph A. (1993). Boundary Layer Analysis. Englewood Cliffs, NJ: Prentice-Hall, Inc. pp. 132–134. ISBN 0-13-086885-X.
15. ^ Fanning friction factor
16. ^ Feigenbaum constants
17. ^ Fresnel number
18. ^ Gain Ratio - Sheldon Brown
19. ^ Incropera, Frank P. (2007). Fundamentals of heat and mass transfer. John Wiley & Sons, Inc. p. 376.
20. ^ Lockhart–Martinelli parameter
21. ^ Manning coefficient PDF (109 KB)
22. ^ Katz J. I. (2009). "When hot water freezes before cold". Am. J. Phys. 77: 27–29. arXiv:physics/0604224. Bibcode:2009AmJPh..77...27K. doi:10.1119/1.2996187. [1] Mpemba number
23. ^ Peel number
24. ^ Richardson number
25. ^ Schmidt number
26. ^ Sommerfeld number
27. ^ Strouhal number, Engineering Toolbox
28. ^ Weaver flame speed number
29. ^ Weissenberg number
30. ^ Womersley number