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Knudsen number
 
 
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The Knudsen number (Kn) is a dimensionless number defined as the ratio
Ratio

A ratio is an expression which compares quantities relative to each other. The most common examples involve two quantities, but in theory any number of quantities can be compared....
 of the molecular mean free path
Mean free path

In physics the mean free path of a particle is the average distance covered by a particle between subsequent impacts....
 length to a representative physical length scale
Scale (ratio)

The concept of scale is applicable if a system is represented Proportionality ly by another system. For example, for a scale model of an object, the ratio of corresponding lengths is a Dimensionless number scale, e.g....
. This length scale could be, for example, the radius
RADIUS

Remote Authentication Dial In User Service is a networking protocol that provides centralized access, authorization and accounting management for people or computers to connect and use a network service....
 of a body in a fluid. The number is named after Danish
Denmark

Denmark is a Scandinavian country in northern Europe and the senior member of the Kingdom of Denmark. It is the southernmost of the Nordic countries....
 physicist Martin Knudsen
Martin Knudsen

This article is about the Danish physicist Martin Knudsen. For the Norwegian footballer, see Martin Knudsen .Martin Hans Christian Knudsen was a Denmark physicist who taught and conducted research at the Technical University of Denmark...
 (1871–1949).

e

(* For particle dynamics in the atmosphere
Atmosphere

An atmosphere is a layer of gases that may surround a material body of sufficient mass, by the gravity of the body, and are retained for a longer duration if gravity is high and the atmosphere's temperature is low....
, and assuming standard temperature and pressure, i.e.






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The Knudsen number (Kn) is a dimensionless number defined as the ratio
Ratio

A ratio is an expression which compares quantities relative to each other. The most common examples involve two quantities, but in theory any number of quantities can be compared....
 of the molecular mean free path
Mean free path

In physics the mean free path of a particle is the average distance covered by a particle between subsequent impacts....
 length to a representative physical length scale
Scale (ratio)

The concept of scale is applicable if a system is represented Proportionality ly by another system. For example, for a scale model of an object, the ratio of corresponding lengths is a Dimensionless number scale, e.g....
. This length scale could be, for example, the radius
RADIUS

Remote Authentication Dial In User Service is a networking protocol that provides centralized access, authorization and accounting management for people or computers to connect and use a network service....
 of a body in a fluid. The number is named after Danish
Denmark

Denmark is a Scandinavian country in northern Europe and the senior member of the Kingdom of Denmark. It is the southernmost of the Nordic countries....
 physicist Martin Knudsen
Martin Knudsen

This article is about the Danish physicist Martin Knudsen. For the Norwegian footballer, see Martin Knudsen .Martin Hans Christian Knudsen was a Denmark physicist who taught and conducted research at the Technical University of Denmark...
 (1871–1949).

Definition


The Knudsen number is defined as:

where

  • = mean free path
    Mean free path

    In physics the mean free path of a particle is the average distance covered by a particle between subsequent impacts....
     (m)*
  • L = representative physical length scale (m)


For an ideal gas
Ideal gas

The ideal gas model is a model of matter in which the molecules are treated as non-interacting point particles which are engaged in a random motion that obeys conservation of energy....
, the mean free path
Mean free path

In physics the mean free path of a particle is the average distance covered by a particle between subsequent impacts....
 may be readily calculated so that:

where
  • kB = Boltzmann's constant
    Boltzmann constant

    The Boltzmann constant is the physical constant relating energy at the particle level with temperature observed at the bulk level. It is the gas constant R divided by the Avogadro constant NA:...
     (approximately 1.38×10-23 J
    Joule

    The joule is the SI derived unit of energy in the International System of Units. It is defined as:One joule is the amount of energy required to perform the following actions:...
    /K
    Kelvin

    The kelvin is a Units of measurement of temperature and is one of the seven SI base units. The Kelvin scale is a Thermodynamic temperature scale where absolute zero, the theoretical absence of all thermal energy, is zero ....
    )
  • T = temperature (K)
  • = particle diameter (m)
  • P = total pressure (Pa
    Pascal (unit)

    The pascal is the SI derived unit of pressure, stress , Young's modulus and tensile strength. It is a measure of force per unit area i.e. equivalent to one newton per square meter or one joule per cubic meter....
    )


(* For particle dynamics in the atmosphere
Atmosphere

An atmosphere is a layer of gases that may surround a material body of sufficient mass, by the gravity of the body, and are retained for a longer duration if gravity is high and the atmosphere's temperature is low....
, and assuming standard temperature and pressure, i.e. 25 °C, 1 atm, we have = 8×10-8 m.)

The Knudsen number is also related to the Mach number
Mach number

Mach number is the speed of an object moving through air, or any fluid substance, divided by the speed of sound as it is in that substance. It is commonly used to represent an object's speed, when it is travelling at the speed of sound....
 and the Reynolds number
Reynolds number

In fluid mechanics and heat transfer, the Reynolds number is a dimensionless number that gives a measure of the ratio of inertial forces to viscosity forces and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions....
 by the following relation:

where is the ratio of specific heats.

Application


The Knudsen number is useful for determining whether statistical mechanics
Statistical mechanics

Statistical mechanics is the application of probability theory, which includes Mathematics tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force....
 or the continuum mechanics
Continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and mechanical behavior of materials modeled as a continuum, e.g., solids and fluids ....
 formulation of fluid dynamics
Fluid dynamics

In physics, fluid dynamics is the sub-discipline of fluid mechanics dealing with fluid flow — the natural science of fluids in motion....
 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
Fluid mechanics

Fluid mechanics is the study of how fluids move and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest, and fluid dynamics, the study of fluids in motion....
 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
Dust

Dust is a general name for minute solid particles with diameters less than 20 Thou . Particles in the Earth's atmosphere arise from various sources such as soil dust lifted up by wind, volcanic eruptions, and pollution....
 particle through the lower atmosphere
Earth's atmosphere

The Earth's atmosphere is a layer of gases surrounding the planet Earth that is retained by the Earth's gravity. Dry air contains roughly 78.08% nitrogen, 20.95% oxygen, 0.93% argon, 0.038% Carbon dioxide in the Earth's atmosphere, and trace amounts of other gases....
, or the motion of a satellite
Satellite

In the context of spaceflight, a satellite is an Physical body which has been placed into orbit by human endeavor. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon....
 through the exosphere
Exosphere

The exosphere is the uppermost layer of an atmosphere. In the exosphere, an upward travelling molecule can escape to space or be pulled back to the celestial body by gravity with little probability of colliding with another molecule....
. The solution of the flow around an aircraft
Aircraft

An aircraft is a vehicle which is able to flight by being supported by the air, or in general, the atmosphere, of a planet. Examples include balloons, airplanes and helicopters....
 has a low Knudsen number. Using the Knudsen number an adjustment for Stokes' Law
Stokes' law

In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force ? also called drag force ? exerted on sphere objects with very small Reynolds numbers in a continuous viscosity fluid....
 can be used in the Cunningham correction factor
Cunningham correction factor

The Cunningham Slip Correction is a correction to the drag coefficient that is used to predict the drag force between a fluid and a particle moving through this fluid....
, this is a drag force correction due to slip in small particles (i.e. dp < 5 µm).

See also

  • Cunningham correction factor
    Cunningham correction factor

    The Cunningham Slip Correction is a correction to the drag coefficient that is used to predict the drag force between a fluid and a particle moving through this fluid....
  • Fluid dynamics
    Fluid dynamics

    In physics, fluid dynamics is the sub-discipline of fluid mechanics dealing with fluid flow — the natural science of fluids in motion....
  • Mach number
    Mach number

    Mach number is the speed of an object moving through air, or any fluid substance, divided by the speed of sound as it is in that substance. It is commonly used to represent an object's speed, when it is travelling at the speed of sound....
  • Reynolds number
    Reynolds number

    In fluid mechanics and heat transfer, the Reynolds number is a dimensionless number that gives a measure of the ratio of inertial forces to viscosity forces and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions....
  • Knudsen Flow
    Knudsen flow

    Knudsen flow describes the movement of fluids with a high Knudsen number, that is, where the characteristic dimension of the flow space is of the same or smaller order of magnitude as the mean free path....