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| | I am Zelda and was born on 16 November 1979. My hobbies are Australian Football League and Juggling.<br><br>Review my web-site - Oltramare ([http://www.moneyhouse.ch/fr/u/fondation_yves_et_inez_oltramare_CH-660.0.989.995-9.htm www.moneyhouse.ch]) |
| The '''Knudsen number''' ('''Kn''') is a [[dimensionless number]] defined as the [[ratio]] of the molecular [[mean free path]] length to a representative physical length [[scale (ratio)|scale]]. This length scale could be, for example, the [[radius]] of the body in a fluid. The number is named after [[Denmark|Danish]] physicist [[Martin Knudsen]] (1871–1949).
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| ==Definition==
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| The Knudsen number is a dimensionless number defined as:
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| :<math>\mathrm{Kn} = \frac {\lambda}{L}</math>
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| where
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| * <math>\lambda</math> = [[mean free path]] [L<sup>1</sup>]
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| * <math>L</math> = representative physical length scale [L<sup>1</sup>].
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| For a [[Boltzmann]] gas, the [[mean free path]] may be readily calculated so that:
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| :<math>\mathrm{Kn} = \frac {k_B T}{\sqrt{2}\pi\sigma^2 p L}</math>
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| where
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| * <math>k_B</math> is the [[Boltzmann constant]] (1.3806504(24) × 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>]
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| * <math>T</math> is the [[thermodynamic temperature]], [θ<sup>1</sup>]
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| * <math>\sigma</math> is the particle hard shell diameter, [L<sup>1</sup>]
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| * <math>p</math> is the total pressure, [M<sup>1</sup> L<sup>-1</sup> T<sup>-2</sup>].
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| For particle dynamics in the [[atmosphere]], and assuming [[standard temperature and pressure]], i.e. 25 °C and 1 atm, we have <math>\lambda</math> ≈ 8 × 10<sup>−8</sup> m.
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| ==Relationship to Mach and Reynolds numbers in gases==
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| The Knudsen number can be related to the [[Mach number]] and the [[Reynolds number]]:
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| Noting the following:
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| [[Dynamic viscosity]],
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| :<math>\mu =\frac{1}{2}\rho \bar{c} \lambda.</math>
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| Average molecule speed (from [[Maxwell-Boltzmann distribution]]),
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| :<math>\bar{c} = \sqrt{\frac{8 k_BT}{\pi m}}</math>
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| thus the mean free path,
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| :<math>\lambda =\frac{\mu }{\rho }\sqrt{\frac{\pi m}{2 k_BT}}</math>
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| dividing through by ''L'' (some characteristic length) the Knudsen number is obtained:
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| :<math>\frac{\lambda }{L}=\frac{\mu }{\rho L}\sqrt{\frac{\pi m}{2 k_BT}}</math>
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| where
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| * ''<math>\bar{c}</math>'' is the average molecular speed from the [[Maxwell–Boltzmann distribution]], [L<sup>1</sup> T<sup>-1</sup>]
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| * ''T'' is the [[thermodynamic temperature]], [θ<sup>1</sup>]
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| * ''μ'' is the [[dynamic viscosity]], [M<sup>1</sup> L<sup>-1</sup> T<sup>-1</sup>]
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| * ''m'' is the [[molecular mass]], [M<sup>1</sup>]
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| * ''k<sub>B</sub>'' is the [[Boltzmann constant]], [M<sup>1</sup> L<sup>2</sup> T<sup>-2</sup> θ<sup>-1</sup>]
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| * ''ρ'' is the density, [M<sup>1</sup> L<sup>-3</sup>].
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| The dimensionless Mach number can be written:
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| :<math>\mathrm{Ma} = \frac {U_\infty}{c_s}</math>
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| where the speed of sound is given by
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| :<math>c_s=\sqrt{\frac{\gamma R T}{M}}=\sqrt{\frac{\gamma k_BT}{m}}</math>
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| where
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| * ''U<sub>∞</sub>'' is the freestream speed, [L<sup>1</sup> T<sup>-1</sup>]
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| * ''R'' is the Universal [[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>]
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| * ''M'' is the [[molar mass]], [M<sup>1</sup> 'mol'<sup>-1</sup>]
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| * <math>\gamma</math> is the [[ratio of specific heats]], and is dimensionless.
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| The dimensionless [[Reynolds number]] can be written:
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| :<math>\mathrm{Re} = \frac {\rho U_\infty L}{\mu}.</math>
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| Dividing the Mach number by the Reynolds number,
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| :<math>\frac{\mathrm{Ma}}{\mathrm{Re}}=\frac{U_\infty / c_s}{\rho U_\infty L / \mu }=\frac{\mu }{\rho L c_s}=\frac{\mu }{\rho L \sqrt{\frac{\gamma k_BT}{m}}}=\frac{\mu }{\rho L }\sqrt{\frac{m}{\gamma k_BT}}</math>
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| and by multiplying by <math>\sqrt{\frac{\gamma \pi }{2}}</math>,
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| :<math>\frac{\mu }{\rho L }\sqrt{\frac{m}{\gamma k_BT}}\sqrt{\frac{\gamma \pi }{2}}=\frac{\mu }{\rho L }\sqrt{\frac{\pi m}{2k_BT}}=\frac{1}{\nu L }\sqrt{\frac{\pi m}{2k_BT}} = \mathrm{Kn}</math>
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| yields the Knudsen number.
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| The Mach, Reynolds and Knudsen numbers are therefore related by:
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| :<math>\mathrm{Kn} = \frac{\mathrm{Ma}}{\mathrm{Re}} \; \sqrt{ \frac{\gamma \pi}{2}}.</math>
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| ==Application==
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| The Knudsen number is useful for determining whether [[statistical mechanics]] or the [[continuum mechanics]] formulation of [[fluid dynamics]] should be used: 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 [[fluid mechanics]] is no longer a good approximation. In this case statistical methods must be used.
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| Problems with high Knudsen numbers include the calculation of the motion of a [[dust]] particle through the lower [[Earth's atmosphere|atmosphere]], or the motion of a [[satellite]] through the [[exosphere]]. One of the most widely used applications for the Knudsen number is in [[microfluidics]] and [[MEMS]] device design. The solution of the flow around an [[aircraft]] has a low Knudsen number, making it firmly in the realm of continuum mechanics. Using the Knudsen number an adjustment for [[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> < 5 µm).
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| ==See also==
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| * [[Cunningham correction factor]]
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| * [[Fluid dynamics]]
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| * [[Mach number]]
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| * [[Knudsen Flow]]
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| * [[Knudsen diffusion]]
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| == References ==
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| * {{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}}
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| {{NonDimFluMech}}
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| [[Category:Dimensionless numbers]]
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| [[Category:Fluid dynamics]]
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I am Zelda and was born on 16 November 1979. My hobbies are Australian Football League and Juggling.
Review my web-site - Oltramare (www.moneyhouse.ch)