List of dimensionless quantities
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
Chemistry
Physics
Physical constants
Fluids and heat transfer
Solids
Optics
Other
Name
Standard symbol
Definition
Named after
Field of application
Fine-structure constant
α
{\displaystyle \alpha }
α
=
e
2
4
π
ε
0
ℏ
c
{\displaystyle \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
{\displaystyle P_{H}}
P
H
=
Z
d
n
d
n
i
{\displaystyle P_{H}={\frac {Z_{d}n_{d}}{n_{i}}}}
O. Havnes
In dusty plasma physics, ratio of the total charge
Z
d
{\displaystyle Z_{d}}
d
{\displaystyle d}
i
{\displaystyle i}
n
{\displaystyle n}
Helmholtz number
H
e
{\displaystyle He}
H
e
=
ω
a
c
0
=
k
0
a
{\displaystyle He={\frac {\omega a}{c_{0}}}=k_{0}a}
Hermann von Helmholtz The most important parameter in duct acoustics . If
ω
{\displaystyle \omega }
frequency , then
k
0
{\displaystyle k_{0}}
wavenumber and
H
e
{\displaystyle He}
[ 7]
Lundquist number S
S
=
μ
0
L
V
A
η
{\displaystyle S={\frac {\mu _{0}LV_{A}}{\eta }}}
Stig Lundqvist
plasma physics (ratio of a resistive time to an Alfvén wave crossing time in a plasma)
Perveance K
K
=
I
I
0
2
β
3
γ
3
(
1
−
γ
2
f
e
)
{\displaystyle {K}={\frac {I}{I_{0}}}\,{\frac {2}{{\beta }^{3}{\gamma }^{3}}}(1-\gamma ^{2}f_{e})}
charged particle transport (measure of the strength of space charge in a charged particle beam)
Pierce parameter
C
{\displaystyle C}
C
3
=
Z
c
I
K
4
V
K
{\displaystyle C^{3}={\frac {Z_{c}I_{K}}{4V_{K}}}}
Traveling wave tube
Beta
β
{\displaystyle \beta }
β
=
n
k
B
T
B
2
/
2
μ
0
{\displaystyle \beta ={\frac {nk_{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
ν
{\displaystyle \nu }
ν
=
−
d
ε
t
r
a
n
s
d
ε
a
x
i
a
l
{\displaystyle \nu =-{\frac {\mathrm {d} \varepsilon _{\mathrm {trans} }}{\mathrm {d} \varepsilon _{\mathrm {axial} }}}}
elasticity (strain in transverse and longitudinal direction)
Q factor Q
Q
=
2
π
f
r
Energy Stored
Power Loss
{\displaystyle 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
R
D
=
ρ
s
u
b
s
t
a
n
c
e
ρ
r
e
f
e
r
e
n
c
e
{\displaystyle 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
μ
r
{\displaystyle \mu _{r}}
μ
r
=
μ
μ
0
{\displaystyle \mu _{r}={\frac {\mu }{\mu _{0}}}}
magnetostatics (ratio of the permeability of a specific medium to free space)
Relative permittivity
ε
r
{\displaystyle \varepsilon _{r}}
ε
r
=
C
x
C
0
{\displaystyle \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
S
t
e
=
c
p
Δ
T
L
{\displaystyle \mathrm {Ste} ={\frac {c_{p}\Delta T}{L}}}
Josef Stefan phase change , thermodynamics (ratio of sensible heat to latent heat )
Strain
ϵ
{\displaystyle \epsilon }
ϵ
=
∂
F
∂
X
−
1
{\displaystyle \epsilon ={\cfrac {\partial {F}}{\partial {X}}}-1}
materials science , elasticity (displacement between particles in the body relative to a reference length)
Mathematics and statistics
Sport
Name
Standard symbol
Definition
Field of application
Blondeau number
B
κ
{\displaystyle B_{\kappa }}
B
κ
=
t
g
v
f
l
m
f
{\displaystyle \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]
Goal difference GD
Goal difference
=
goals scored
−
goals conceded
{\displaystyle {\text{Goal difference}}={\text{goals scored}}-{\text{goals conceded}}}
Association football [ 11]
Runs Per Wicket Ratio RpW ratio
RpW ratio
=
runs scored
wickets lost
÷
runs conceded
wickets taken
{\displaystyle {\text{RpW ratio }}={\frac {\text{runs scored}}{\text{wickets lost}}}\div {\frac {\text{runs conceded}}{\text{wickets taken}}}}
cricket [ 12]
Winning percentage –
Various, e.g.
Games won
Games played
{\displaystyle {\frac {\text{Games won}}{\text{Games played}}}}
Points won
Points contested
{\displaystyle {\frac {\text{Points won}}{\text{Points contested}}}}
Various sports
Other fields
Name
Standard symbol
Definition
Field of application
Capacity factor
actual electrical energy output
maximum possible electrical energy output
{\displaystyle {\frac {\text{actual electrical energy output}}{\text{maximum possible electrical energy output}}}}
energy
Cohesion number
Coh
C
o
h
=
1
ρ
g
(
Γ
5
E
∗
2
R
∗
8
)
1
3
{\displaystyle 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)[ 13] [ 14]
Cost of transport COT
C
O
T
=
E
m
g
d
{\displaystyle \mathrm {COT} ={\frac {E}{mgd}}}
energy efficiency , economics (ratio of energy input to kinetic motion)
Damping ratio
ζ
{\displaystyle \zeta }
ζ
=
c
2
k
m
{\displaystyle \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
=
∂
ln
(
x
)
∂
ln
(
y
)
=
∂
x
∂
y
y
x
{\displaystyle 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
average load
peak load
{\displaystyle {\frac {\text{average load}}{\text{peak load}}}}
energy
Peel number N P
N
P
=
Restoring force
Adhesive force
{\displaystyle N_{\mathrm {P} }={\frac {\text{Restoring force}}{\text{Adhesive force}}}}
coating (adhesion of microstructures with substrate )[ 15]
Pixel px
digital imaging (smallest addressable unit)
Power factor pf
p
f
=
P
S
{\displaystyle pf={\frac {P}{S}}}
electrical (real power to apparent power)
Power number N p
N
p
=
P
ρ
n
3
d
5
{\displaystyle N_{p}={P \over \rho n^{3}d^{5}}}
fluid mechanics, power consumption by rotary agitators ; resistance force versus inertia force)
Prater number β
β
=
−
Δ
H
r
D
T
A
e
C
A
S
λ
e
T
s
{\displaystyle \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)[ 16]
Relative density RD
R
D
=
ρ
s
u
b
s
t
a
n
c
e
ρ
r
e
f
e
r
e
n
c
e
{\displaystyle RD={\frac {\rho _{\mathrm {substance} }}{\rho _{\mathrm {reference} }}}}
hydrometers , material comparisons (ratio of density of a material to a reference material—usually water )
References
^ "Table of Dimensionless Numbers" (PDF) . 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 376 . ISBN 9780470055540 ^ 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 : 228171. 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 ^ "goal difference" . Cambridge Dictionary .^ "World Test Championship Playing Conditions: What's different?" (PDF) . 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" (PDF) . 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
