The Full Wiki

Knudsen number: Wikis

Advertisements
  
  

Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.

Encyclopedia

From Wikipedia, the free encyclopedia

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

The Knudsen number is a dimensionless number defined as:

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

where

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

For an ideal gas, the mean free path may be readily calculated so that:

\mathit{Kn} = \frac {k_B T}{\sqrt{2}\pi\sigma^2 p L}

where

  • kB 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]
  • σ 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 λ ≈ 8 × 10−8 m, or approximately 2.6 × 10−9 ft.

Relationship to Mach and Reynolds numbers

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

Noting the following:

Dynamic viscosity,

\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:

\mathit{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

The dimensionless Reynolds number can be written:

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

Dividing the Mach number by the Reynolds number,

\frac{Ma}{Re}=\frac{U_\infty \div c_s}{\rho U_\infty 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}}

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}}

the Knudsen number is obtained.

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

Kn = \frac{Ma}{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. 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

Advertisements

Advertisements






Got something to say? Make a comment.
Your name
Your email address
Message