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:
<math display="block">\mathrm{Kn}\ = \frac {\lambda}{L},</math>
where:
The representative length scale considered, <math>L</math>, may correspond to various physical traits of a system, but most commonly relates to a gap length over which thermal transport or mass transport occurs through a gas phase. This is the case in porous and granular materials, where the thermal transport through a gas phase depends highly on its pressure and the consequent mean free path of molecules in this phase. For a Boltzmann gas, the mean free path may be readily calculated, so that:
<math display="block">\mathrm{Kn}\ = \frac {k_\text{B} T}{\sqrt{2}\pi d^2 p L}=\frac {k_\text{B{\sqrt{2}\pi d^2 \rho R_{s} L},</math>
where:
If the temperature is increased, but the volume kept constant, then the Knudsen number (and the mean free path) doesn't change (for an ideal gas). In this case, the density stays the same. If the temperature is increased, and the pressure kept constant, then the gas expands and therefore its density decreases. In this case, the mean free path increases and so does the Knudsen number. Hence, it may be helpful to keep in mind that the mean free path (and therefore the Knudsen number) is really dependent on the thermodynamic variable density (proportional to the reciprocal of density), and only indirectly on temperature and pressure.
For particle dynamics in the atmosphere, and assuming standard temperature and pressure, i.e. 0 °C and 1 atm, we have <math>\lambda</math> ≈ (80 nm).
Relationship to Mach and Reynolds numbers in gases
The Knudsen number can be related to the Mach number and the Reynolds number.
Using the dynamic viscosity:
<math display="block">\mu = \frac{1}{2}\rho \bar{c} \lambda,</math>
with the average molecule speed (from Maxwell–Boltzmann distribution):
<math display="block">\bar{c} = \sqrt{\frac{8 k_\text{B} T}{\pi m,</math>
the mean free path is determined as follows:
<math display="block">\lambda = \frac{\mu}{\rho} \sqrt{\frac{\pi m}{2 k_\text{B} T.</math>
Dividing through by L (some characteristic length), the Knudsen number is obtained:
<math display="block"> \mathrm{Kn}\ = \frac{\lambda}{L} = \frac{\mu}{\rho L} \sqrt{\frac{\pi m}{2 k_\text{B} T,</math>
where:
The dimensionless Mach number can be written as:
<math display="block">\mathrm{Ma} = \frac {U_\infty}{c_\text{s,</math>
where the speed of sound is given by:
<math display="block">c_\text{s} = \sqrt{\frac{\gamma R T}{M = \sqrt{\frac{\gamma k_\text{B}T}{m,</math>
where:
The dimensionless Reynolds number can be written as:
<math display="block">\mathrm{Re} = \frac {\rho U_\infty L}{\mu}.</math>
Dividing the Mach number by the Reynolds number:
<math display="block">\frac{\mathrm{Ma{\mathrm{Re = \frac{U_\infty / c_\text{s{\rho U_\infty L / \mu} = \frac{\mu}{\rho L c_\text{s = \frac{\mu}{\rho L \sqrt{\frac{\gamma k_\text{B} T}{m} = \frac{\mu}{\rho L} \sqrt{\frac{m}{\gamma k_\text{B} T</math>
and by multiplying by <math>\sqrt{\frac{\gamma \pi}{2</math> yields the Knudsen number:
<math display="block">\frac{\mu}{\rho L} \sqrt{\frac{m}{\gamma k_\text{B}T \sqrt{\frac{\gamma \pi}{2 = \frac{\mu}{\rho L} \sqrt{\frac{\pi m}{2k_\text{B} T = \mathrm{Kn}.</math>
The Mach, Reynolds and Knudsen numbers are therefore related by:
<math display="block">\mathrm{Kn}\ = \frac{\mathrm{Ma{\mathrm{Re \sqrt{\frac{\gamma \pi}{2.</math>
Application
The Knudsen number can be used to determine the rarefaction of a flow:
- <math>\mathrm{Kn} < 0.01 </math>: Continuum flow
- <math>0.01 < \mathrm{Kn} < 0.1 </math>: Slip flow
- <math> 0.1 < \mathrm{Kn} < 10 </math>: Transitional flow
- <math>\mathrm{Kn} > 10 </math>: Free molecular flow
This regime classification is empirical and problem dependent but has proven useful to adequately model flows.
Problems with high Knudsen numbers include the calculation of the motion of a dust particle through the lower atmosphere and 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 where flows range from continuum to free-molecular. It has also been successfully demonstrated for use in hydrogen production from water.
The Knudsen number also plays an important role in thermal conduction in gases. For insulation materials, for example, where gases are contained under low pressure, the Knudsen number should be as high as possible to ensure low thermal conductivity.
See also
References
External links
- Knudsen number and diffusivity calculators