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1/13/13 Knudsen number - Wikipedia, the free encyclopedia en.wikipedia.org/wiki/Knudsen_number 1/4 Knudsen number 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 the body in a fluid. The number is named after Danish physicist Martin Knudsen (1871–1949). Contents 1 Definition 2 Relationship to Mach and Reynolds numbers in gases 3 Application 4 See also 5 References Definition The Knudsen number is a dimensionless number defined as: where = mean free path [L 1 ] = representative physical length scale [L 1 ]. For an ideal gas, the mean free path may be readily calculated so that: where is the Boltzmann constant (1.3806504(24) × 10 −23 J/K in SI units), [M 1 L 2 T -2 θ -1 ] is the thermodynamic temperature, [θ 1 ] is the particle hard shell diameter, [L 1 ] is the total pressure, [M 1 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 ≈ 8 × 10 −8 m.
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1/13/13 Knudsen number - Wikipedia, the free encyclopedia
en.wikipedia.org/wiki/Knudsen_number 1/4
Knudsen numberFrom Wikipedia, the free encyclopedia
The Knudsen number (Kn) is a dimensionless number defined as the ratio of the molecular mean free path lengthto 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).
Contents
1 Definition
2 Relationship to Mach and Reynolds numbers in gases
3 Application
4 See also
5 References
Definition
The Knudsen number is a dimensionless number defined as:
where
= mean free path [L1]
= representative physical length scale [L1].
For an ideal gas, the mean free path may be readily calculated so that:
where
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].
For particle dynamics in the atmosphere, and assuming standard temperature and pressure, i.e. 25 °C and 1 atm,
we have ≈ 8 × 10−8 m.
1/13/13 Knudsen number - Wikipedia, the free encyclopedia
en.wikipedia.org/wiki/Knudsen_number 2/4
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,
Average molecule speed (from Maxwell-Boltzmann distribution),
thus the mean free path,
dividing through by L (some characteristic length) the Knudsen number is obtained:
where
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
where
1/13/13 Knudsen number - Wikipedia, the free encyclopedia
en.wikipedia.org/wiki/Knudsen_number 3/4
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:
Application
The Knudsen number is useful for determining whether statistical mechanics or the continuum mechanics formulationof fluid dynamics should be used: If the Knudsen number is near or greater than one, the mean free path of amolecule is comparable to a length scale of the problem, and the continuum assumption of fluid mechanics is nolonger 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 loweratmosphere, or the motion of a satellite through the exosphere. One of the most widely used applications for theKnudsen number is in microfluidics and MEMS device design. The solution of the flow around an aircraft has a lowKnudsen number, making it firmly in the realm of continuum mechanics. Using the Knudsen number an adjustmentfor Stokes' Law can be used in the Cunningham correction factor, this is a drag force correction due to slip in smallparticles (i.e. dp < 5 µm).
See also
1/13/13 Knudsen number - Wikipedia, the free encyclopedia
en.wikipedia.org/wiki/Knudsen_number 4/4
Cunningham correction factor
Fluid dynamics
Mach numberKnudsen Flow
Knudsen diffusion
References
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