#jsDisabledContent { display:none; } My Account |  Register |  Help

# Knudsen number

Article Id: WHEBN0000401314
Reproduction Date:

 Title: Knudsen number Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Knudsen number

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).

## Contents

• Definition 1
• Relationship to Mach and Reynolds numbers in gases 2
• Application 3
• References 5

## Definition

The Knudsen number is a dimensionless number defined as:

\mathrm{Kn} = \frac {\lambda}{L}

where

• \lambda = mean free path [L1]
• L = representative physical length scale [L1].

For a Boltzmann gas, the mean free path may be readily calculated so that:

\mathrm{Kn} = \frac {k_B T}{\sqrt{2}\pi d^2 p L}

where

• k_B is the Boltzmann constant (1.3806504(24) × 10−23 J/K in SI units), [M1 L2 T−2 θ−1]
• T is the thermodynamic temperature, [θ1]
• d is the particle hard shell diameter, [L1]
• p is the total pressure, [M1 L−1 T−2].

For particle dynamics in the atmosphere, and assuming standard temperature and pressure, i.e. 25 °C and 1 atm, we have \lambda ≈ 8 × 10−8 m.

## Relationship to Mach and Reynolds numbers in gases

The Knudsen number can be related to the Mach number and the Reynolds number:

Noting the following:

\mu =\frac{1}{2}\rho \bar{c} \lambda.

Average molecule speed (from Maxwell–Boltzmann distribution),

\bar{c} = \sqrt{\frac{8 k_BT}{\pi m}}

thus the mean free path,

\lambda =\frac{\mu }{\rho }\sqrt{\frac{\pi m}{2 k_BT}}

dividing through by L (some characteristic length) the Knudsen number is obtained:

\frac{\lambda }{L}=\frac{\mu }{\rho L}\sqrt{\frac{\pi m}{2 k_BT}}

where

The dimensionless Mach number can be written:

\mathrm{Ma} = \frac {U_\infty}{c_s}

where the speed of sound is given by

c_s=\sqrt{\frac{\gamma R T}{M}}=\sqrt{\frac{\gamma k_BT}{m}}

where

• U is the freestream speed, [L1 T−1]
• R is the Universal gas constant, (in SI, 8.314 47215 J K−1 mol−1), [M1 L2 T−2 θ−1 'mol'−1]
• M is the molar mass, [M1 'mol'−1]
• \gamma is the ratio of specific heats, and is dimensionless.

The dimensionless Reynolds number can be written:

\mathrm{Re} = \frac {\rho U_\infty L}{\mu}.

Dividing the Mach number by the Reynolds number,

\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}}

and by multiplying by \sqrt{\frac{\gamma \pi }{2}},

\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}} = \mathrm{Kn}

yields the Knudsen number.

The Mach, Reynolds and Knudsen numbers are therefore related by:

\mathrm{Kn} = \frac{\mathrm{Ma}}{\mathrm{Re}} \; \sqrt{ \frac{\gamma \pi}{2}}.

## 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. 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. dp < 5 µm).

## References

• Cussler, E. L. (1997). Diffusion: Mass Transfer in Fluid Systems. Cambridge University Press.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.