list of dimensionless quantities

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This is a list of well-known dimensionless quantities illustrating their variety of forms and applications. The tables also include pure numbers, dimensionless ratios, or dimensionless physical constants; these topics are discussed in the article.

Biology and medicine

scope="col" \| Name ! scope="col" \| Standard symbol ! scope="col" class="unsortable" \| Definition ! scope="col" \| Field of application
class="wikitable sortable"
Basic reproduction number	$R_0$	number of infections caused on average by an infectious individual over entire infectious period	epidemiology
Body fat percentage		total mass of fat divided by total body mass, multiplied by 100	biology
Kt/V	Kt/V		medicine (hemodialysis and peritoneal dialysis treatment; dimensionless time)
Waist–hip ratio		waist circumference divided by hip circumference	biology
Waist-to-chest ratio		waist circumference divided by chest circumference	biology
Waist-to-height ratio		waist circumference divided by height	biology

Chemistry

scope="col" \| Name ! scope="col" \| Standard symbol ! scope="col" class="unsortable" \| Definition !Named after ! scope="col" \| Field of application
class="wikitable sortable"
Activity coefficient	$\gamma$	$\gamma= \frac {{a}}{{x}}$ \|	chemistry (Proportion of "active" molecules or atoms)
Arrhenius number	$\alpha$	$\alpha = \frac{E_a}{RT}$ \|Svante Arrhenius	chemistry (ratio of activation energy to thermal energy){{cite web \|title=Table of Dimensionless Numbers \|url=http://www.cchem.berkeley.edu/gsac/grad_info/prelims/binders/dimensionless_numbers.pdf \|access-date=2009-11-05}}
Atomic weight	M	\|	chemistry (mass of one atom divided by the atomic mass constant, {{val\|1\|ul=Da}})
Bodenstein number	Bo or Bd	$\mathrm{Bo} = vL/\mathcal{D} = \mathrm{Re}\, \mathrm{Sc}$ \|Max Bodenstein	chemistry (residence-time distribution; similar to the axial mass transfer Peclet number){{Cite journal \| last1 = Becker \| first1 = A. \| last2 = Hüttinger \| first2 = K. J. \| doi = 10.1016/S0008-6223(97)00175-9 \| title = Chemistry and kinetics of chemical vapor deposition of pyrocarbon—II pyrocarbon deposition from ethylene, acetylene and 1,3-butadiene in the low temperature regime \| journal = Carbon \| volume = 36 \| issue = 3 \| pages = 177 \| year = 1998 }}
Damköhler numbers	Da	$\mathrm{Da} = k \tau$ \|Gerhard Damköhler	chemistry (reaction time scales vs. residence time)
Hatta number	Ha	$\mathrm{Ha} = \frac{N_{\mathrm{A}0}}{N_{\mathrm{A}0}^{\mathrm{phys}}}$ \|Shirôji Hatta (1895–1973)	chemical engineering (adsorption enhancement due to chemical reaction)
Jakob number	Ja	$\mathrm{Ja} = \frac{c_p (T_\mathrm{s} - T_\mathrm{sat}) }{\Delta H_{\mathrm{f}} }$ \|	chemistry (ratio of sensible to latent energy absorbed during liquid-vapor phase change){{cite book \|last=Incropera \|first=Frank P. \|title= Fundamentals of heat and mass transfer \|url=https://archive.org/details/fundamentalsheat00incr_869 \|url-access=limited \|page=[https://archive.org/details/fundamentalsheat00incr_869/page/n383 376] \|year=2007 \|publisher=John Wiley & Sons, Inc\|isbn=9780470055540 }}
pH	$\mathrm{pH}$	$\mathrm{pH} = - \log_{10}(a_{\textrm{H}^+})$ \|	chemistry (the measure of the acidity or basicity of an aqueous solution)
van 't Hoff factor	i	$i = 1 + \alpha (n - 1)$ \|Jacobus Henricus van 't Hoff	quantitative analysis (K_f and K_b)
Wagner number	Wa	$\mathrm{Wa} = \frac{\kappa}{l} \frac{\mathrm{d}\eta}{\mathrm{d}i}$ \|	electrochemistry (ratio of kinetic polarization resistance to solution ohmic resistance in an electrochemical cell){{cite book \|last1=Popov \|first1=Konstantin I. \|last2=Djokić \|first2=Stojan S. \|last3=Grgur \|first3=Branimir N. \|title=Fundamental Aspects of Electrometallurgy \|date=2002 \|publisher=Springer \|location=Boston, MA \|isbn=978-0-306-47564-1 \|pages=101–102}}
Weaver flame speed number	Wea	$\mathrm{Wea} = \frac{w}{w_\mathrm{H}} 100$ \|	combustion (laminar burning velocity relative to hydrogen gas){{cite book\| last1 = Kuneš \| first1 = J.\| chapter = Technology and Mechanical Engineering\| doi = 10.1016/B978-0-12-416013-2.00008-7\| title = Dimensionless Physical Quantities in Science and Engineering\| pages = 353–390\| year = 2012\| isbn = 978-0-12-416013-2}}

Physics

=Physical constants=

{{main|Dimensionless physical constant#Examples}}

=Fluids and heat transfer=

{{main|Dimensionless numbers in fluid mechanics}}

=Solids=

scope="col" \| Name ! scope="col" \| Standard symbol ! scope="col" class="unsortable" \| Definition !Named after ! scope="col" \| Field of application
class="wikitable sortable"
Coefficient of kinetic friction	$\mu_k$	\|	mechanics (friction of solid bodies in translational motion)
Coefficient of static friction	$\mu_s$	\|	mechanics (friction of solid bodies at rest)
Föppl–von Kármán number	$\gamma$	$\gamma = \frac{Y r^2}{\kappa}$ \|August Föppl and Theodore von Kármán	virology, solid mechanics (thin-shell buckling)
Rockwell scale	–	\|Hugh M. (1890–1957) and Stanley P. (1886–1940) Rockwell	mechanical hardness (indentation hardness of a material)
Rolling resistance coefficient	C_rr	$C_{rr} = \frac{F}{N_f}$ \|	vehicle dynamics (ratio of force needed for motion of a wheel over the normal force)

=Optics=

scope="col" \| Name ! scope="col" \| Standard symbol ! scope="col" class="unsortable" \| Definition !Named after ! scope="col" \| Field of application
class="wikitable sortable"
Abbe number	V	$V = \frac{ n_d - 1 }{ n_F - n_C }$ \|Ernst Abbe	optics (dispersion in optical materials)
f-number	N	$N = \frac{f}{D}$ \|	optics, photography (ratio of focal length to diameter of aperture)
Fresnel number	F	$\mathit{F} = \frac{a^{2}}{L \lambda}$ \|Augustin-Jean Fresnel	optics (slit diffraction)[http://www.ilt.fraunhofer.de/default.php?web=1&id=100050&lan=eng&dat=2 Fresnel number] {{webarchive\|url=https://web.archive.org/web/20111001052854/http://www.ilt.fraunhofer.de/default.php?web=1&id=100050&lan=eng&dat=2 \|date=2011-10-01 }}
Refractive index	n	$n=\frac{c}{v}$ \|	electromagnetism, optics (speed of light in vacuum over speed of light in a material)
Transmittance	T	$T = \frac{I}{I_0}$ \|	optics, spectroscopy (the ratio of the intensities of radiation exiting through and incident on a sample)

= Other =

scope="col" \| Name ! scope="col" \| Standard symbol ! scope="col" class="unsortable" \| Definition !Named after ! scope="col" \| Field of application
class="wikitable sortable"
Fine-structure constant	$\alpha$	$\alpha = \frac{e^2}{4\pi\varepsilon_0 \hbar c}$ \|	quantum electrodynamics (QED) (coupling constant characterizing the strength of the electromagnetic interaction)
Havnes parameter	$P_H$	$P_H = \frac{Z_d n_d}{n_i}$ \|O. Havnes	In dusty plasma physics, ratio of the total charge $Z_d$ carried by the dust particles $d$ to the charge carried by the ions $i$ , with $n$ the number density of particles
Helmholtz number	$He$	$He = \frac{\omega a}{c_0} = k_0a$ \|Hermann von Helmholtz	The most important parameter in duct acoustics. If $\omega$ is the dimensional frequency, then $k_0$ is the corresponding free field wavenumber and $He$ is the corresponding dimensionless frequency S.W. RIENSTRA, 2015, Fundamentals of Duct Acoustics, Von Karman Institute Lecture Notes
Lundquist number	S	$S = \frac{\mu_0LV_A}{\eta}$ \|Stig Lundqvist	plasma physics (ratio of a resistive time to an Alfvén wave crossing time in a plasma)
Perveance	K	${K} = \frac{{I}}{{I_0}}\,\frac{{2}}{{\beta}^3{\gamma}^3} (1-\gamma^2f_e)$ \|	charged particle transport (measure of the strength of space charge in a charged particle beam)
Pierce parameter	$C$	$C^3=\frac{Z_c I_K}{4 V_K}$ \|	Traveling wave tube
Beta	$\beta$	$\beta = \frac{n k_B T}{B^2/2\mu_0}$ \|	Plasma and fusion power. Ratio of plasma thermal pressure to magnetic pressure, controlling the level of turbulence in a magnetised plasma.
Poisson's ratio	$\nu$	$\nu = -\frac{\mathrm{d}\varepsilon_\mathrm{trans}}{\mathrm{d}\varepsilon_\mathrm{axial}}$ \|	elasticity (strain in transverse and longitudinal direction)
Q factor	Q	$Q = 2 \pi f_r \frac{\text{Energy Stored}}{\text{Power Loss}}$ \|	physics, engineering (Damping ratio of oscillator or resonator; energy stored versus energy lost)
Relative density	RD	$RD = \frac{\rho_\mathrm{substance}}{\rho_\mathrm{reference}}$ \|	hydrometers, material comparisons (ratio of density of a material to a reference material—usually water)
Relative permeability	$\mu_r$	$\mu_r = \frac{\mu}{\mu_0}$ \|	magnetostatics (ratio of the permeability of a specific medium to free space)
Relative permittivity	$\varepsilon_r$	$\varepsilon_{r} = \frac{C_{x}} {C_{0}}$ \|	electrostatics (ratio of capacitance of test capacitor with dielectric material versus vacuum)
Specific gravity	SG	\|	(same as Relative density)
Stefan number	Ste	$\mathrm{Ste} = \frac{c_p \Delta T}{L}$ \|Josef Stefan	phase change, thermodynamics (ratio of sensible heat to latent heat)
Strain	$\epsilon$	$\epsilon = \cfrac{\partial{F}}{\partial{X}} - 1$ \|	materials science, elasticity (displacement between particles in the body relative to a reference length)
Erlang	$E$	$E = \lambda h$ \| Agner Krarup Erlang	telephony (a measure of offered load on a telephone circuit)

Mathematics and statistics

{{Main|List of mathematical constants}}

Geography, geology and geophysics

scope="col" \| Name ! scope="col" \| Standard symbol ! scope="col" class="unsortable" \| Definition !Named after ! scope="col" \| Field of application
class="wikitable sortable"
Albedo	$\alpha$	$\alpha= (1-D) \bar \alpha(\theta_i) + D \bar{ \bar \alpha}$ \|	climatology, astronomy (reflectivity of surfaces or bodies)
Dieterich–Ruina–Rice number \| $\mathrm{R_u}$ \| $\mathrm{R_u} = \frac{W}{L}\frac{(b-a)\bar{\sigma}}{G}$ \|James H. Dieterich, Andy Ruina, and James R. Rice \|mechanics, friction, rheology, geophysics (stiffness ratio for frictional contacts){{Cite journal \|last1=Barbot \|first1=S. \|year=2019 \|title=Slow-slip, slow earthquakes, period-two cycles, full and partial ruptures, and deterministic chaos in a single asperity fault \|journal=Tectonophysics \|volume=768 \|pages=228171 \|bibcode=2019Tectp.76828171B \|doi=10.1016/j.tecto.2019.228171 \|doi-access=free}}
Love numbers	h, k, l	\|Augustus Edward Hough Love	geophysics (solidity of earth and other planets)
Porosity	$\phi$	$\phi = \frac{V_\mathrm{V}}{V_\mathrm{T}}$ \|	geology, porous media (void fraction of the medium)
Rossby number	Ro	$\mathrm{Ro}=\frac{U}{Lf}$ \|Carl-Gustav Arvid Rossby	geophysics (ratio of inertial to Coriolis force)

Sport

scope="col" \| Name ! scope="col" \| Standard symbol ! scope="col" class="unsortable" \| Definition ! scope="col" \| Field of application
class="wikitable sortable"
Blondeau number	$B_\kappa$	$\mathrm{B_\kappa} = \frac{t_g v_f}{l_{mf}}$	sport science, team sports{{cite journal \|last1=Blondeau \|first1=J. \|title=The influence of field size, goal size and number of players on the average number of goals scored per game in variants of football and hockey: the Pi-theorem applied to team sports \|journal=Journal of Quantitative Analysis in Sports \|year=2021 \|volume=17 \|issue=2 \|pages=145–154 \|doi=10.1515/jqas-2020-0009 \|s2cid=224929098 \|url=https://doi.org/10.1515/jqas-2020-0009}}
Gain ratio	–		bicycling (system of representing gearing; length traveled over length pedaled)[http://sheldonbrown.com/gain.html Gain Ratio – Sheldon Brown]
Runs Per Wicket Ratio	RpW ratio	$\text{RpW ratio }=\frac{\text{runs scored}}{\text{wickets lost}} \div \frac{\text{runs conceded}}{\text{wickets taken}}$	cricket{{Cite web\|url=https://icc-static-files.s3.amazonaws.com/ICC/document/2019/07/31/6b4241d8-1b33-44b5-8a83-579380989fb9/Changes-to-Test-PCs-for-WTC.pdf\|title=World Test Championship Playing Conditions: What's different?\|last=\|first=\|date=\|website=International Cricket Council\|archive-url=\|archive-date=\|access-date=11 August 2021}}
Winning percentage	–	Various, e.g. $\frac{\text{Games won}}{\text{Games played}}$ or $\frac{\text{Points won}}{\text{Points contested}}$	Various sports

Other fields

scope="col" \| Name ! scope="col" \| Standard symbol ! scope="col" class="unsortable" \| Definition ! scope="col" \| Field of application
class="wikitable sortable"
Capacity factor		$\frac{\text{actual electrical energy output}}{\text{maximum possible electrical energy output}}$	energy
Cohesion number \|Coh \| $Coh=\frac{1}{\rho g}\left ( \frac{\Gamma^5}{{E^}^2{R^}^8} \right )^{\frac{1}{3}}$ \|Chemical engineering, material science, mechanics (A scale to show the energy needed for detaching two solid particles){{Cite journal\|last1=Behjani\|first1=Mohammadreza Alizadeh\|last2=Rahmanian\|first2=Nejat\|last3=Ghani\|first3=Nur Fardina bt Abdul\|last4=Hassanpour\|first4=Ali\|title=An investigation on process of seeded granulation in a continuous drum granulator using DEM\|journal=Advanced Powder Technology\|volume=28\|issue=10\|pages=2456–2464\|doi=10.1016/j.apt.2017.02.011\|year=2017\|url=http://eprints.whiterose.ac.uk/113300/3/APT_Seeded_Granulation-accepted%20manuscript%20Feb%202017.pdf}}{{Cite journal\|last1=Alizadeh Behjani\|first1=Mohammadreza\|last2=Hassanpour\|first2=Ali\|last3=Ghadiri\|first3=Mojtaba\|last4=Bayly\|first4=Andrew\|date=2017\|title=Numerical Analysis of the Effect of Particle Shape and Adhesion on the Segregation of Powder Mixtures\|journal=EPJ Web of Conferences\|language=en\|volume=140\|pages=06024\|doi=10.1051/epjconf/201714006024\|issn=2100-014X\|bibcode=2017EPJWC.14006024A\|doi-access=free}}
Cost of transport	COT	$\mathrm{COT} = \frac{E}{mgd}$	energy efficiency, economics (ratio of energy input to kinetic motion)
Damping ratio	$\zeta$	$\zeta = \frac{c}{2 \sqrt{km}}$	mechanics, electrical engineering (the level of damping in a system)
Decibel	dB		acoustics, electronics, control theory (ratio of two intensities or powers of a wave)
Elasticity (economics)	E	$E_{x,y} = \frac{\partial \ln(x)}{\partial \ln(y)} = \frac{\partial x}{\partial y}\frac{y}{x}$	economics (response of demand or supply to price changes)
Gain	–		electronics (signal output to signal input)
Load factor		$\frac{\text{average load}}{\text{peak load}}$	energy
Peel number	N_P	$N_\mathrm{P} = \frac{\text{Restoring force}}{\text{Adhesive force}}$	coating (adhesion of microstructures with substrate){{Cite journal \| last1 = Van Spengen \| first1 = W. M. \| last2 = Puers \| first2 = R. \| last3 = De Wolf \| first3 = I. \| doi = 10.1109/TDMR.2003.820295 \| title = The prediction of stiction failures in MEMS \| journal = IEEE Transactions on Device and Materials Reliability \| volume = 3 \| issue = 4 \| pages = 167 \| year = 2003 }}
Pixel	px		digital imaging (smallest addressable unit)
Power factor	pf	$pf = \frac{P}{S}$ \| electrical (real power to apparent power)
Power number	N_p	$N_p = {P\over \rho n^3 d^5}$	fluid mechanics, power consumption by rotary agitators; resistance force versus inertia force)
Prater number	β	$\beta = \frac{-\Delta H_r D_{TA}^e C_{AS}}{\lambda^e T_s}$	reaction engineering (ratio of heat evolution to heat conduction within a catalyst pellet){{cite book \|last1=Davis \|first1=Mark E. \|last2=Davis \|first2=Robert J. \|title=Fundamentals of Chemical Reaction Engineering \|year=2012 \|publisher=Dover \|isbn=978-0-486-48855-4 \|page=215}}
Relative density	RD	$RD = \frac{\rho_\mathrm{substance}}{\rho_\mathrm{reference}}$	hydrometers, material comparisons (ratio of density of a material to a reference material—usually water)

References

{{Reflist|30em}}

Bibliography

{{cite web | website=iso.org |title=ISO 80000-11:2019 Quantities and units — Part 11: Characteristic numbers | url=https://www.iso.org/obp/ui/#iso:std:iso:80000:-11:ed-2:v1:en | access-date=2023-08-31}}