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The Knudsen number |
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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:<math>\mu =\frac{1}{2}\rho \bar{c} \lambda</math> |
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:<math>\bar{c} = \sqrt{\frac{8 k_BT}{\pi m}}</math> |
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thus, <math>\lambda =\frac{\mu }{\rho }\sqrt{\frac{\pi m}{2 k_BT}}</math> |
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and 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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noting ''c<sub>s</sub>'' is the average molecular speed from the [[Maxwell–Boltzmann distribution]], ''T'' is the absolute temperature, ''μ'' is the [[dynamic viscosity]], ''m'' is the [[molecular mass]] and ''k<sub>B</sub>'' is the [[Boltzmann constant]]. |
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The Mach number can be written: |
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:<math>\mathit{Ma} = \frac {U}{c_s}</math> |
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where the speed of sound is given by <math>c_s=\sqrt{\frac{\gamma R T}{M}}=\sqrt{\frac{\gamma k_BT}{m}}</math> |
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noting ''U'' is the speed of the body (possessing characteristic length ''L'') relative to the gas in question, ''R'' is the Universal [[gas constant]], ''M'' is the [[molar mass]] and <math>\gamma</math> is the [[ratio of specific heats]]. |
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The [[Reynolds number]] can be written: |
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:<math>\mathit{Re} = \frac {\mu V L}{\rho}</math> |
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Dividing the Mach number by the Reynolds number, |
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:<math>\frac{Ma}{Re}=\frac{U \div c_s}{\rho U L \div \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}}</math> |
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the Knudsen number is obtained. The Mach, Reynolds and Knudsen numbers are therefore related by: |
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:<math> |
:<math> |
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Kn = \frac{Ma}{Re} \; \sqrt{ \frac{\ |
Kn = \frac{Ma}{Re} \; \sqrt{ \frac{\gamma \pi}{2}} |
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</math> |
</math> |
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where <math>\gamma</math> is the [[ratio of specific heats]]. |
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==Application== |
==Application== |
Revision as of 15:39, 27 May 2009
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. This length scale could be, for example, the radius of a body in a fluid. The number is named after Danish physicist Martin Knudsen (1871–1949).
Definition
The Knudsen number is defined as:
where
- = mean free path (m)*
- = representative physical length scale (m)
For an ideal gas, the mean free path may be readily calculated so that:
where
- = Boltzmann's constant (approximately 1.38 × 10−23 J/K)
- = temperature (K)
- = particle diameter (m)
- = total pressure (Pa)
(* For particle dynamics in the atmosphere, and assuming standard temperature and pressure, i.e. 25 °C, 1 atm, we have = 8 × 10−8 m.)
The Knudsen number can be related to the Mach number and the Reynolds number:
Noting the following:
thus,
and dividing through by L (some characteristic length) the Knudsen number is obtained:
noting cs is the average molecular speed from the Maxwell–Boltzmann distribution, T is the absolute temperature, μ is the dynamic viscosity, m is the molecular mass and kB is the Boltzmann constant.
The Mach number can be written:
where the speed of sound is given by
noting U is the speed of the body (possessing characteristic length L) relative to the gas in question, R is the Universal gas constant, M is the molar mass and is the ratio of specific heats.
The Reynolds number can be written:
Dividing the Mach number by the Reynolds number,
and by multiplying by ,
the Knudsen number is obtained. The Mach, Reynolds and Knudsen numbers are therefore related by:
Application
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.
Problems with high Knudsen numbers include the calculation of the motion of a dust particle through the lower atmosphere, or the motion of a satellite through the exosphere. The solution of the flow around an aircraft has a low Knudsen number. 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. dp < 5 µm).
See also
References