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).

The Knudsen number helps determine whether statistical mechanics or the continuum mechanics formulation of fluid dynamics should be used to model a situation. 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 such cases, statistical methods should be used.

Definition

The Knudsen number is a dimensionless number defined as

${\displaystyle \mathrm {Kn} ={\frac {\lambda }{L}},}$

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

${\displaystyle \mathrm {Kn} ={\frac {k_{B}T}{{\sqrt {2}}\pi d^{2}pL}},}$

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 (80 nm).

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),

${\bar {c}}={\sqrt {{\frac {8k_{B}T}{\pi m}}}}$

thus the mean free path,

$\lambda ={\frac {\mu }{\rho }}{\sqrt {{\frac {\pi m}{2k_{B}T}}}}$

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

${\frac {\lambda }{L}}={\frac {\mu }{\rho L}}{\sqrt {{\frac {\pi m}{2k_{B}T}}}}$

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:

${\mathrm {Ma}}={\frac {U_{\infty }}{c_{s}}}$

where the speed of sound is given by

$c_{s}={\sqrt {{\frac {\gamma RT}{M}}}}={\sqrt {{\frac {\gamma k_{B}T}{m}}}}$

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,

${\frac {{\mathrm {Ma}}}{{\mathrm {Re}}}}={\frac {U_{\infty }/c_{s}}{\rho U_{\infty }L/\mu }}={\frac {\mu }{\rho Lc_{s}}}={\frac {\mu }{\rho L{\sqrt {{\frac {\gamma k_{B}T}{m}}}}}}={\frac {\mu }{\rho L}}{\sqrt {{\frac {m}{\gamma k_{B}T}}}}$

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

${\frac {\mu }{\rho L}}{\sqrt {{\frac {m}{\gamma k_{B}T}}}}{\sqrt {{\frac {\gamma \pi }{2}}}}={\frac {\mu }{\rho L}}{\sqrt {{\frac {\pi m}{2k_{B}T}}}}={\mathrm {Kn}}$

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

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. Movements of fluids in situations with a high Knudsen number are said to exhibit Knudsen flow.

Airflow 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). The flow of water through a nozzle will usually be a situation with a low Knudsen number.^[1]

Mixtures of gases with different molecular masses can be partly separated by sending the mixture through small holes of a thin wall because the numbers of molecules that pass through a hole is proportional to the pressure of the gas and inversely proportional to its molecular mass. The technique has been used to separate isotopic mixtures, such as uranium, using porous membranes,^[2] It has also been successfully demonstrated for use in hydrogen production from water.^[3]

One source says that Kn>10 is a suitable criterion for distinguishing molecular flow from continuum flow.^[1]

See also

• Cunningham correction factor
• Fluid dynamics
• Mach number
• Knudsen flow
• Knudsen diffusion
• Knudsen paradox

References

1. Laurendeau, Normand M. (2005). Statistical thermodynamics: fundamentals and applications. Cambridge University Press. p. 306. ISBN 0-521-84635-8., Appendix N, page 434
2. Villani, S. (1976). Isotope Separation. Hinsdale, Ill.: American Nuclear Society.
3. Kogan, A. (1998). "Direct solar thermal splitting of water and on-site separation of the products - II. Experimental feasibility study". Int. J. Hydrogen Energy. Great Britain: Elsevier Science Ltd. 23 (2): 89–98. doi:10.1016/S0360-3199(97)00038-4.

• Cussler, E. L. (1997). Diffusion: Mass Transfer in Fluid Systems. Cambridge University Press. ISBN 0-521-45078-0.

External links

• Knudsen number and diffusivity calculators