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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).
The Knudsen number is a dimensionless number defined as:
- = mean free path [L1]
- = representative physical length scale [L1].
- is the Boltzmann constant (1.3806504(24) × 10−23 J/K in SI units), [M1 L2 T−2 θ−1]
- is the thermodynamic temperature, [θ1]
- is the particle hard shell diameter, [L1]
- is the total pressure, [M1 L−1 T−2].
Relationship to Mach and Reynolds numbers in gases
Noting the following:
Average molecule speed (from Maxwell–Boltzmann distribution),
thus the mean free path,
dividing through by L (some characteristic length) the Knudsen number is obtained:
- 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]
- kB is the Boltzmann constant, [M1 L2 T−2 θ−1]
- ρ is the density, [M1 L−3].
The dimensionless Mach number can be written:
where the speed of sound is given by
- 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]
- is the ratio of specific heats, and is dimensionless.
The dimensionless Reynolds number can be written:
Dividing the Mach number by the Reynolds number,
and by multiplying by ,
yields the Knudsen number.
The Mach, Reynolds and Knudsen numbers are therefore related by:
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).
- Cussler, E. L. (1997). Diffusion: Mass Transfer in Fluid Systems. Cambridge University Press. ISBN 0-521-45078-0.