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{{Short description|Dimensionless number relating to mean free path of a particle}}
The '''Knudsen number''' ('''Kn''') is a dimensionless number defined as the [[Ratio|ratio]] of the molecular [[Physics:Mean free path|mean free path]] length to a representative physical length scale. This length scale could be, for example, the [[Radius|radius]] of a body in a fluid. The number is named after {{wipe|Denmark}} physicist [[Biography:Martin Knudsen|Martin Knudsen]] (1871–1949).

The Knudsen number helps determine whether [[Physics:Statistical mechanics|statistical mechanics]] or the [[Physics:Continuum mechanics|continuum mechanics]] formulation of [[Physics:Fluid dynamics|fluid dynamics]] should be used to model a situation. If the Knudsen number is near or greater than one, the mean free path of a molecule is comparable to a length scale of the problem, and the continuum assumption of [[Physics:Fluid mechanics|fluid mechanics]] is no longer a good approximation.  In such cases, statistical methods should be used.

==Definition==
The Knudsen number is a dimensionless number defined as

:<math>\mathrm{Kn}\ = \frac {\lambda}{L},</math>

where
: <math>\lambda</math> = [[Physics:Mean free path|mean free path]] [L<sup>1</sup>],
: <math>L</math> = representative physical length scale [L<sup>1</sup>].

The representative length scale considered, <math>L</math>, may correspond to various physical traits of a system, but most commonly relates to a ''gap length'' over which thermal transport or mass transport occurs through a gas phase. This is the case in porous and granular materials, where the thermal transport through a gas phase depends highly on its pressure and the consequent mean free path of molecules in this phase.<ref>{{cite journal| last1=Dai  |display-authors=0 | title= Effective Thermal Conductivity of Submicron Powders: A Numerical Study| journal= Applied Mechanics and Materials| year=2016 | volume=846| pages=500–505| url=https://www.researchgate.net/publication/305644421 |doi=10.4028/www.scientific.net/AMM.846.500 |s2cid=114611104 }}</ref> For a Boltzmann gas, the [[Physics:Mean free path|mean free path]] may be readily calculated, so that

:<math>\mathrm{Kn}\ = \frac {k_\text{B} T}{\sqrt{2}\pi d^2 p L}=\frac {k_\text{B}}{\sqrt{2}\pi d^2 \rho R_{s} L},</math>

where
: <math>k_\text{B}</math> is the [[Physics:Boltzmann constant|Boltzmann constant]] (1.380649 × 10<sup>−23</sup> J/K in SI units) [M<sup>1</sup> L<sup>2</sup> T<sup>−2</sup> Θ<sup>−1</sup>],
: <math>T</math> is the [[Physics:Thermodynamic temperature|thermodynamic temperature]] [θ<sup>1</sup>],
: <math>d</math> is the particle hard-shell diameter [L<sup>1</sup>],
: <math>p</math> is the static pressure [M<sup>1</sup> L<sup>−1</sup> T<sup>−2</sup>],
: <math>R_{s}</math> is the [[Physics:Gas constant#Specific gas constant|specific gas constant]] [L<sup>2</sup> T<sup>−2</sup> θ<sup>−1</sup>] (287.05 J/(kg K) for air),
: <math>\rho</math> is the density [M<sup>1</sup> L<sup>−3</sup>].

If the temperature is increased, but the ''volume'' kept constant, then the Knudsen number (and the mean free path) doesn't change (for an ideal gas). In this case, the density stays the same. If the temperature is increased, and the ''pressure'' kept constant, then the gas expands and therefore its density decreases. In this case, the mean free path increases and so does the Knudsen number. Hence, it may be helpful to keep in mind that the mean free path (and therefore the Knudsen number) is really dependent on the thermodynamic variable density (proportional to the reciprocal of density), and only indirectly on temperature and pressure.

For particle dynamics in the [[Astronomy:Atmosphere|atmosphere]], and assuming [[Standard temperature and pressure|standard temperature and pressure]], i.e. 0&nbsp;°C and 1 atm, we have <math>\lambda</math> ≈ {{val|8e-8|u=m}} (80&nbsp;nm).

==Relationship to Mach and Reynolds numbers in gases==
The Knudsen number can be related to the [[Physics:Mach number|Mach number]] and the [[Reynolds number]].

Using the [[Physics:Dynamic viscosity|dynamic viscosity]]
:<math>\mu = \frac{1}{2}\rho \bar{c} \lambda,</math>

with the average molecule speed (from [[Maxwell–Boltzmann distribution]])
:<math>\bar{c} = \sqrt{\frac{8 k_\text{B} T}{\pi m}},</math>

the [[Physics:Mean free path|mean free path]] is determined as follows:<ref name = "thermal">{{cite journal | last1= Dai | first1= W. |display-authors=0 | title= Influence of gas pressure on the effective thermal conductivity of ceramic breeder pebble beds | journal = Fusion Engineering and Design | year=2017 | volume=118| pages= 45–51|doi= 10.1016/j.fusengdes.2017.03.073 }}</ref>
:<math>\lambda = \frac{\mu}{\rho} \sqrt{\frac{\pi m}{2 k_\text{B} T}}.</math>

Dividing through by ''L'' (some characteristic length), the Knudsen number is obtained:
:<math> \mathrm{Kn}\ = \frac{\lambda}{L} = \frac{\mu}{\rho L} \sqrt{\frac{\pi m}{2 k_\text{B} T}},</math>

where
: <math>\bar{c}</math> is the average molecular speed from the [[Maxwell–Boltzmann distribution]] [L<sup>1</sup> T<sup>−1</sup>],
: ''T'' is the [[Physics:Thermodynamic temperature|thermodynamic temperature]] [θ<sup>1</sup>],
: ''μ'' is the [[Physics:Dynamic viscosity|dynamic viscosity]] [M<sup>1</sup> L<sup>−1</sup> T<sup>−1</sup>],
: ''m'' is the [[Physics:Molecular mass|molecular mass]] [M<sup>1</sup>],
: ''k<sub>B</sub>'' is the [[Physics:Boltzmann constant|Boltzmann constant]] [M<sup>1</sup> L<sup>2</sup> T<sup>−2</sup> θ<sup>−1</sup>],
: <math>\rho</math> is the density [M<sup>1</sup> L<sup>−3</sup>].

The dimensionless Mach number can be written as
:<math>\mathrm{Ma} = \frac {U_\infty}{c_\text{s}},</math>

where the speed of sound is given by
:<math>c_\text{s} = \sqrt{\frac{\gamma R T}{M}} = \sqrt{\frac{\gamma k_\text{B}T}{m}},</math>

where
: ''U<sub>∞</sub>'' is the freestream speed [L<sup>1</sup> T<sup>−1</sup>],
: ''R'' is the Universal [[Physics:Gas constant|gas constant]] (in SI, 8.314 47215 J K<sup>−1</sup> mol<sup>−1</sup>) [M<sup>1</sup> L<sup>2</sup> T<sup>−2</sup> θ<sup>−1</sup> mol<sup>−1</sup>],
: ''M'' is the [[Physics:Molar mass|molar mass]] [M<sup>1</sup> mol<sup>−1</sup>],
: <math>\gamma</math> is the ratio of specific heats [1].

The dimensionless [[Reynolds number]] can be written as
:<math>\mathrm{Re} = \frac {\rho U_\infty L}{\mu}.</math>

Dividing the Mach number by the Reynolds number:

:<math>\frac{\mathrm{Ma}}{\mathrm{Re}} = \frac{U_\infty / c_\text{s}}{\rho U_\infty L / \mu} = \frac{\mu}{\rho L c_\text{s}} = \frac{\mu}{\rho L \sqrt{\frac{\gamma k_\text{B} T}{m}}} = \frac{\mu}{\rho L} \sqrt{\frac{m}{\gamma k_\text{B} T}}</math>

and by multiplying by <math>\sqrt{\frac{\gamma \pi}{2}}</math> yields the Knudsen number:

:<math>\frac{\mu}{\rho L} \sqrt{\frac{m}{\gamma k_\text{B}T}} \sqrt{\frac{\gamma \pi}{2}} = \frac{\mu}{\rho L} \sqrt{\frac{\pi m}{2k_\text{B} T}} = \mathrm{Kn}.</math>

The Mach, Reynolds and Knudsen numbers are therefore related by

:<math>\mathrm{Kn}\ = \frac{\mathrm{Ma}}{\mathrm{Re}} \sqrt{\frac{\gamma \pi}{2}}.</math>

==Application==

The Knudsen number can be used to determine the rarefaction of a flow:<ref name="karniadakis-00">{{cite book|title=Microflows and nanoflows: fundamentals and simulation|year=2000|publisher=Springer|author=Karniadakis, G. and Beskok, A. and Aluru, N.}}</ref>
<ref name="Ziarani-00">{{cite conference|last=Ziarani A. S.|first=Aguilera R., Cui X. C.|title= Permeability of Tight Sand and Shale Formations: A Dual Mechanism Approach for Micro and Nanodarcy Reservoirs|publisher=SPE |year=2020|isbn=978-1-61399-685-0|conference=SPE Canada Unconventional Resources Conference. SPE-200010-MS}}</ref>
* <math>\mathrm{Kn} < 0.01 </math>: [[Physics:Continuum mechanics|Continuum flow]]
* <math>0.01 < \mathrm{Kn} < 0.1 </math>:  Slip flow
* <math> 0.1 < \mathrm{Kn} < 10 </math>:  Transitional flow
* <math>\mathrm{Kn} > 10 </math>: [[Physics:Free molecular flow|Free molecular flow]]<ref name=Laurendeau>{{cite book
|title=Statistical thermodynamics: fundamentals and applications
|first1=Normand M.
|last1=Laurendeau
|publisher=Cambridge University Press
|year=2005
|isbn=0-521-84635-8
|page=306
|url=https://books.google.com/books?id=QF6iMewh4KMC}}, [https://books.google.com/books?id=QF6iMewh4KMC&pg=PA434 Appendix N, page 434]
</ref>
This regime classification is empirical and problem dependent but has proven useful to adequately model flows.<ref name="karniadakis-00" />
<ref name="Cussler-00">{{cite book|last=Cussler|first=E. L.|title=Diffusion: Mass Transfer in Fluid Systems|publisher=Cambridge University Press|year=1997|isbn=0-521-45078-0}}</ref>

Problems with high Knudsen numbers include the calculation of the motion of a dust particle through the lower atmosphere and the motion of a [[Astronomy:Satellite|satellite]] through the [[Earth:Exosphere|exosphere]]. One of the most widely used applications for the Knudsen number is in [[Physics:Microfluidics|microfluidics]] and [[Engineering:MEMS|MEMS]] device design where flows range from continuum to free-molecular.<ref name="karniadakis-00" /> In recent years, it has been applied in other disciplines such as transport in porous media, e.g., petroleum reservoirs.<ref name="Ziarani-00"/> Movements of fluids in situations with a high Knudsen number are said to exhibit [[Physics:Knudsen flow|Knudsen flow]], also called [[Physics:Free molecular flow|free molecular flow]].

Airflow around an [[Engineering:Aircraft|aircraft]] such as an airliner has a low Knudsen number, making it firmly in the realm of continuum mechanics. Using the Knudsen number an adjustment for [[Physics:Stokes' law|Stokes' law]] can be used in the [[Cunningham correction factor]], this is a drag force correction due to slip in small particles (i.e. ''d''<sub>''p''</sub>&nbsp;< 5&nbsp;μm). The flow of water through a nozzle will usually be a situation with a low Knudsen number.<ref name=Laurendeau/>

Mixtures of gases with different molecular masses can be partly separated by sending the mixture through small holes of a thin wall because the numbers of molecules that pass through a hole is proportional to the pressure of the gas and inversely proportional to its molecular mass. The technique has been used to separate [[Physics:Isotope|isotopic]] mixtures, such as [[Chemistry:Uranium|uranium]], using porous membranes,<ref>{{cite book | last = Villani | first = S. | title = Isotope Separation | publisher = American Nuclear Society | date = 1976 | location = Hinsdale, Ill.}}</ref> It has also been successfully demonstrated for use in [[Chemistry:Hydrogen production|hydrogen production]] from water.<ref>{{cite journal | doi = 10.1016/S0360-3199(97)00038-4 | last = Kogan | first = A. | title = Direct solar thermal splitting of water and on-site separation of the products - II. Experimental feasibility study | journal = International Journal of Hydrogen Energy | volume = 23 | issue = 2 | pages = 89–98 | publisher = Elsevier Science Ltd | location = Great Britain | date = 1998}}</ref>

The Knudsen number also plays an important role in thermal conduction in gases. For insulation materials, for example, where gases are contained under low pressure, the Knudsen number should be as high as possible to ensure low [[Physics:Thermal conductivity|thermal conductivity]].<ref>{{Cite web|url=https://www.tec-science.com/thermodynamics/heat/thermal-conductivity-of-gases/|title=Thermal conductivity of gases|last=tec-science|date=2020-01-27|website=tec-science|language=en-US|access-date=2020-03-22}}</ref>

==See also==
* {{annotated link|Cunningham correction factor}}
* {{annotated link|Physics:Fluid dynamics}}
* {{annotated link|Physics:Mach number}}
* {{annotated link|Physics:Free molecular flow|aka=Knudsen flow}}
* {{annotated link|Physics:Knudsen diffusion}}
* {{annotated link|Knudsen paradox}}

== References ==

{{Reflist}}

==External links==
* [http://www.fxsolver.com/solve/share/mP2Benn1q2W-whj98QSwRQ==/ Knudsen number and diffusivity calculators]

{{NonDimFluMech}}

[[Category:Dimensionless numbers]]
[[Category:Fluid dynamics]]
[[Category:Dimensionless numbers of fluid mechanics]]

{{Sourceattribution|Knudsen number}}