Knudsen number

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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).

[edit] 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

$k B$ 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]
$σ$ 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.

[edit] 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

c_s is the average molecular speed from the Maxwell–Boltzmann distribution, [L¹ T^-1]
T is the thermodynamic temperature, [θ¹]
μ is the dynamic viscosity, [M¹ L^-1 T^-1]
m is the molecular mass, [M¹]
k_B is the Boltzmann constant, [M¹ L² T^-2 θ^-1]
ρ is the density, [M¹ L^-3]

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

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]
$γ$ is the ratio of specific heats, and is dimensionless.

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:

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

[edit] 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. d_p < 5 µm).

[edit] See also

[edit] References

Knudsen number

From Wikipedia, the free encyclopedia

Contents

[edit] Definition

[edit] Relationship to Mach and Reynolds numbers

[edit] Application

[edit] See also

[edit] References

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