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 the body in a fluid. The number is named after Danish physicist Martin Knudsen (1871–1949).

Definition

The Knudsen number is a dimensionless number defined as: \[\mathit{Kn} = \frac {\lambda}{L}\]

where

• \(\lambda\) = 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]
• \(\sigma\) 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:

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

• \(\bar{c}\) 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]
• \(\gamma\) 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 \(\sqrt{\frac{\gamma \pi }{2}}\),

\[\frac{\mu }{\rho L }\sqrt{\frac{m}{\gamma k_BT}}\sqrt{\frac{\gamma \pi }{2}}=\frac{\mu }{\rho L }\sqrt{\frac{\pi m}{2k_BT}}\]how can be equal to that ? \[\mathit{Kn} = \frac {k_B T}{\sqrt{2}\pi\sigma^2 p L}\]

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. 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. d_p < 5 µm).

See also

• Cunningham correction factor
• Fluid dynamics
• Mach number
• Knudsen Flow
• Knudsen diffusion

References

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