Knudsen number
Encyclopedia
The Knudsen number 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).
where
For an ideal gas
, the mean free path
may be readily calculated so that:
where
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.
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
The dimensionless Mach number can be written:
where the speed of sound is given by
where
The dimensionless Reynolds number can be written:
Dividing the Mach number by the Reynolds number,
and by multiplying by ,
the Knudsen number is obtained.
The Mach, Reynolds and Knudsen numbers are therefore related by:
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).
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
of the molecular mean free path
Mean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...
length to a representative physical length scale
Scale (ratio)
The scale ratio of some sort of model which represents an original proportionally is the ratio of a linear dimension of the model to the same dimension of the original. Examples include a 3-dimensional scale model of a building or the scale drawings of the elevations or plans of a building. In such...
. This length scale could be, for example, the radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
of a body in a fluid. The number is named after Danish
Denmark
Denmark is a Scandinavian country in Northern Europe. The countries of Denmark and Greenland, as well as the Faroe Islands, constitute the Kingdom of Denmark . It is the southernmost of the Nordic countries, southwest of Sweden and south of Norway, and bordered to the south by Germany. Denmark...
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 Danish physicist who taught and conducted research at the Technical University of DenmarkHe is primarily known for his study of molecular gas flow and the...
(1871–1949).
Definition
The Knudsen number is a dimensionless number defined as:where
- = mean free pathMean free pathIn physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...
[L^{1}] - = representative physical length scale [L^{1}].
For an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
, the mean free path
Mean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...
may be readily calculated so that:
where
- is the Boltzmann constant (1.3806504(24) × 10^{−23} J/K in SISiSi, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
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
Atmosphere
An atmosphere is a layer of gases that may surround a material body of sufficient mass, and that is held in place by the gravity of the body. An atmosphere may be retained for a longer duration, if the gravity is high and the atmosphere's temperature is low...
, and assuming standard temperature and pressure, i.e. 25 °C and 1 atm, we have ≈ 8 × 10^{−8} m.
Relationship to Mach and Reynolds numbers in Gases
The Knudsen number can be related to the Mach numberMach number
Mach number is the speed of an object moving through air, or any other fluid substance, divided by the speed of sound as it is in that substance for its particular physical conditions, including those of temperature and pressure...
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, [L^{1} T^{-1}]
- T is the thermodynamic temperature, [θ^{1}]
- μ is the dynamic viscosity, [M^{1} L^{-1} T^{-1}]
- m is the molecular massMolecular massThe molecular mass of a substance is the mass of one molecule of that substance, in unified atomic mass unit u...
, [M^{1}] - k_{B} is the Boltzmann constant, [M^{1} L^{2} T^{-2} θ^{-1}]
- ρ is the density, [M^{1} L^{-3}].
The dimensionless Mach number can be written:
where the speed of sound is given by
where
- U_{∞} is the freestream speed, [L^{1} T^{-1}]
- R is the Universal gas constantGas constantThe gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...
, (in SISiSi, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
, 8.314 47215 J K^{−1} mol^{−1}), [M^{1} L^{2} T^{-2} θ^{-1} 'mol'^{-1}] - M is the molar massMolar massMolar mass, symbol M, is a physical property of a given substance , namely its mass per amount of substance. The base SI unit for mass is the kilogram and that for amount of substance is the mole. Thus, the derived unit for molar mass is kg/mol...
, [M^{1} '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 ,
the Knudsen number is obtained.
The Mach, Reynolds and Knudsen numbers are therefore related by:
Application
The Knudsen number is useful for determining whether statistical mechanicsStatistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
or the continuum mechanics
Continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...
formulation of fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
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 fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid 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 consists of particles in the atmosphere that 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 atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...
, or the motion of a satellite
Satellite
In the context of spaceflight, a satellite is an object which has been placed into orbit by human endeavour. 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 Earth's atmosphere. In the exosphere, an upward travelling molecule moving fast enough to attain escape velocity can escape to space with a low chance of collisions; if it is moving below escape velocity it will be prevented from escaping from the celestial...
. One of the most widely used applications for the Knudsen number is in microfluidics
Microfluidics
Microfluidics deals with the behavior, precise control and manipulation of fluids that are geometrically constrained to a small, typically sub-millimeter, scale.Typically, micro means one of the following features:* small volumes...
and MEMS device design. The solution of the flow around an aircraft
Aircraft
An aircraft is a vehicle that is able to fly by gaining support from the air, or, in general, the atmosphere of a planet. An aircraft counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines.Although...
has a low Knudsen number, making it firmly in the realm of continuum mechanics. 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 spherical objects with very small Reynolds numbers in a continuous viscous fluid...
can be used in the Cunningham correction factor
Cunningham correction factor
The Cunningham correction factor or Cunningham slip correction factor is used to account for noncontinuum effects when calculating the drag on small particles. The derivation of Stokes Law, which is used to calculate the drag force on small particles, assumes a No-slip condition which is no longer...
, this is a drag force correction due to slip in small particles (i.e. d_{p} < 5 µm).
See also
- Cunningham correction factorCunningham correction factorThe Cunningham correction factor or Cunningham slip correction factor is used to account for noncontinuum effects when calculating the drag on small particles. The derivation of Stokes Law, which is used to calculate the drag force on small particles, assumes a No-slip condition which is no longer...
- Fluid dynamicsFluid dynamicsIn physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
- Mach numberMach numberMach number is the speed of an object moving through air, or any other fluid substance, divided by the speed of sound as it is in that substance for its particular physical conditions, including those of temperature and pressure...
- Knudsen FlowKnudsen flowKnudsen 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.-Separation processes using Knudsen Flow:...
- Knudsen diffusionKnudsen diffusionKnudsen diffusion is a means of diffusion that occurs in a long pore with a narrow diameter because molecules frequently collide with the pore wall.Consider the diffusion of gas molecules through very small capillary pores...