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
| Name | Standard symbol | Definition | Field of application |
|---|---|---|---|
| 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
| Name | Standard symbol | Definition | Named after | Field of application |
|---|---|---|---|---|
| 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)[1] |
| Atomic weight | M | chemistry (mass of one atom divided by the atomic mass constant, 1 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)[2] |
| 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)[3] | |
| 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 (Kf and Kb) |
| 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)[4] | |
| Weaver flame speed number | Wea | \mathrm{Wea} = \frac{w}{w_\mathrm{H}} 100 | combustion (laminar burning velocity relative to hydrogen gas)[5] |
Physics
Physical constants
Fluids and heat transfer
Solids
| Name | Standard symbol | Definition | Named after | Field of application |
|---|---|---|---|---|
| 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 | Crr | C_{rr} = \frac{F}{N_f} | vehicle dynamics (ratio of force needed for motion of a wheel over the normal force) |
Optics
| Name | Standard symbol | Definition | Named after | Field of application |
|---|---|---|---|---|
| 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)[6] |
| 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
| Name | Standard symbol | Definition | Named after | Field of application |
|---|---|---|---|---|
| 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 [7] |
| 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
Geography, geology and geophysics
| Name | Standard symbol | Definition | Named after | Field of application |
|---|---|---|---|---|
| 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)[8] |
| 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
| Name | Standard symbol | Definition | Field of application |
|---|---|---|---|
| Blondeau number | B_\kappa | \mathrm{B_\kappa} = \frac{t_g v_f}{l_{mf}} | sport science, team sports[9] |
| Gain ratio | – | bicycling (system of representing gearing; length traveled over length pedaled)[10] | |
| 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[11] |
| Winning percentage | – | Various, e.g. \frac{\text{Games won}}{\text{Games played}} or \frac{\text{Points won}}{\text{Points contested}} | Various sports |
Other fields
| Name | Standard symbol | Definition | Field of application |
|---|---|---|---|
| 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)[12][13] |
| 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 | NP | N_\mathrm{P} = \frac{\text{Restoring force}}{\text{Adhesive force}} | coating (adhesion of microstructures with substrate)[14] |
| Pixel | px | digital imaging (smallest addressable unit) | |
| Power factor | pf | pf = \frac{P}{S} | electrical (real power to apparent power) |
| Power number | Np | 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)[15] |
| 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) |
| Drag Coefficient | CD | CD = 2D/\rho v^2 A | Aerodynamics, fluid dynamics, hydrodynamics |
References
- ^ "Table of Dimensionless Numbers". Retrieved 2009-11-05.
- ^ Becker, A. & Hüttinger, K. J. (1998). "Chemistry and kinetics of chemical vapor deposition of pyrocarbon—II pyrocarbon deposition from ethylene, acetylene and 1,3-butadiene in the low temperature regime". Carbon. 36 (3): 177. doi:10.1016/S0008-6223(97)00175-9
- ^ Incropera, Frank P. (2007). Fundamentals of heat and mass transfer. John Wiley & Sons, Inc. p. 376. ISBN 978-0-470-05554-0.
- ^ Popov, Konstantin I.; Djokić, Stojan S.; Grgur, Branimir N. (2002). Fundamental Aspects of Electrometallurgy. Boston, MA: Springer. pp. 101–102. ISBN 978-0-306-47564-1.
- ^ Kuneš, J. (2012). "Technology and Mechanical Engineering". Dimensionless Physical Quantities in Science and Engineering. pp. 353–390. doi:10.1016/B978-0-12-416013-2.00008-7. ISBN 978-0-12-416013-2.
- ^ Fresnel number Archived 2011-10-01 at the Wayback Machine
- ^ S.W. RIENSTRA, 2015, Fundamentals of Duct Acoustics, Von Karman Institute Lecture Notes
- ^ Barbot, S. (2019). "Slow-slip, slow earthquakes, period-two cycles, full and partial ruptures, and deterministic chaos in a single asperity fault". Tectonophysics. 768. Bibcode:2019Tectp.76828171B. doi:10.1016/j.tecto.2019.228171
- ^ Blondeau, J. (2021). "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 of Quantitative Analysis in Sports. 17 (2): 145–154. doi:10.1515/jqas-2020-0009. S2CID 224929098
- ^ Gain Ratio – Sheldon Brown
- ^ "World Test Championship Playing Conditions: What's different?". International Cricket Council. Retrieved 11 August 2021.
- ^ Behjani, Mohammadreza Alizadeh; Rahmanian, Nejat; Ghani, Nur Fardina bt Abdul; Hassanpour, Ali (2017). "An investigation on process of seeded granulation in a continuous drum granulator using DEM". Advanced Powder Technology. 28 (10): 2456–2464. doi:10.1016/j.apt.2017.02.011
- ^ Alizadeh Behjani, Mohammadreza; Hassanpour, Ali; Ghadiri, Mojtaba; Bayly, Andrew (2017). "Numerical Analysis of the Effect of Particle Shape and Adhesion on the Segregation of Powder Mixtures". EPJ Web of Conferences. 140: 06024. Bibcode:2017EPJWC.14006024A. doi:10.1051/epjconf/201714006024. ISSN 2100-014X
- ^ Van Spengen, W. M.; Puers, R.; De Wolf, I. (2003). "The prediction of stiction failures in MEMS". IEEE Transactions on Device and Materials Reliability. 3 (4): 167. doi:10.1109/TDMR.2003.820295
- ^ Davis, Mark E. & Davis, Robert J. (2012). Fundamentals of Chemical Reaction Engineering. Dover. p. 215. ISBN 978-0-486-48855-4.
Bibliography
- "ISO 80000-11:2019 Quantities and units — Part 11: Characteristic numbers". iso.org. Retrieved 2023-08-31.