Knudsen number

Physical key figure
Surname	Knudsen number
Formula symbol	${\displaystyle {\mathit {Kn}}}$
dimension	dimensionless
definition	${\displaystyle {\mathit {Kn}}={\frac {\lambda }{l}}}$
${\displaystyle \lambda }$ free path ${\displaystyle l}$ characteristic length
Named after	Martin Knudsen

The Knudsen number (after the Danish scientist Martin Knudsen ) is a dimensionless number for estimating the flow behavior of a gas flow. It shows the ratio of the mean free path of the gas molecules to a characteristic length of the flow field (e.g. diameter of a pipe through which the flow passes). It shows whether the movement of a gas can be described as a fluid- mechanical continuum or as the movement of individual particles. ${\displaystyle {\mathit {Kn}}}$ ${\displaystyle \lambda }$ ${\displaystyle l}$

${\displaystyle {\mathit {Kn}}={\frac {\lambda }{l}}}$

For an ideal gas that satisfies the Maxwell-Boltzmann distribution , the following applies

${\displaystyle {\mathit {Kn}}={\frac {k_{\mathrm {B} }\cdot T}{{\sqrt {2}}\cdot \pi \cdot \sigma ^{2}\cdot p\cdot l}}}$

With

• ${\displaystyle l}$ : characteristic length of the flow
• ${\displaystyle k_{\mathrm {B} }}$ : Boltzmann's constant (1.38 · 10 ⁻²³  J  ·  K ⁻¹ )
• ${\displaystyle T}$ : Temperature (in K)
• ${\displaystyle \sigma }$ : Diameter of the molecules
• ${\displaystyle p}$ : absolute pressure

For that apply kinetic theory of gases laws of very dilute media (close to vacuum ), while for the laws of gas dynamics of continuous media apply. ${\displaystyle {\mathit {Kn}}\gg 1}$ ${\displaystyle {\mathit {Kn}}\ll 1}$

10	${\displaystyle \leq }$	${\displaystyle {\mathit {Kn}}}$			free molecular flow ( Knudsen gas )
0.1	${\displaystyle \leq }$	${\displaystyle {\mathit {Kn}}}$	${\displaystyle \leq }$	10	Transitional or Knudsen flow ( ) ${\displaystyle \lambda \approx l}$
0.01	${\displaystyle \leq }$	${\displaystyle {\mathit {Kn}}}$	${\displaystyle \leq }$	0.1	Sliding flow
		${\displaystyle {\mathit {Kn}}}$	${\displaystyle \leq }$	0.01	Continuum flow

The Knudsen number also plays an important role in heat conduction in gases. For insulation materials, for example, in which gases are enclosed under low pressure, the Knudsen number should be selected as large as possible in order to ensure low thermal conductivity.

literature

• Dieter Hänel: Molecular Gas Dynamics . Springer, Berlin 2004, ISBN 3-540-44247-2 .

Individual evidence

1. ↑ Heinz M. Hiersig (Ed.): Lexicon of engineering knowledge basics . Springer Verlag, 2013, ISBN 978-3-642-95765-9 , pp. 371 ( limited preview in Google Book search).
2. ↑ tec-science: Thermal conductivity of gases. In: tec-science. January 11, 2020, accessed on March 22, 2020 (German).