Tru Physics

Knudsen Number

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Introduction

The Knudsen number (K_n) is a dimensionless number in fluid dynamics which describes the relative importance of molecular diffusion over molecular momentum transfer in a gas. It is named after the Danish physicist Martin Knudsen.

Definition

The Knudsen number is defined as the ratio of the molecular mean free path \lambda to a characteristic physical length scale L of the problem. This can be expressed mathematically as:

K_n = \dfrac{\lambda}{L}

Mean Free Path

The mean free path of a particle, such as a molecule in a gas, is the average distance that a particle travels between successive collisions. In an ideal gas, the mean free path \lambda can be given by:

\lambda = \dfrac{k_BT}{\sqrt{2} \pi d^2 P}

where k_B is the Boltzmann constant, T is the temperature, d is the molecular diameter, and P is the pressure.

Interpretation

The Knudsen number indicates the degree to which a flow can be treated as a continuum. A Knudsen number much less than one (Kn \ll 1) implies that the flow can be treated as a continuum. If the Knudsen number is of the order of or greater than one (Kn \geq 1), the flow must be treated as a rarefied gas, and the methods of statistical mechanics must be used.

Knudsen Regimes

Different regimes of flow are categorized based on the Knudsen number:

  1. Continuum flow regime (Kn < 0.01)
  2. Slip flow regime (0.01 \leq Kn < 0.1)
  3. Transition flow regime (0.1 \leq Kn < 10)
  4. Free molecular flow regime (Kn \geq 10)

Knudsen Diffusion

Knudsen diffusion occurs when the Knudsen number is much greater than one. In this regime, the gas molecules interact more frequently with the walls of a porous medium than with each other. The Knudsen diffusion coefficient is given by:

D_K = \dfrac{2}{3} \dfrac{\lambda}{\tau} v_{\text{mean}}

where v_{\text{mean}} is the mean molecular velocity and \tau is the porosity of the medium.

Conclusion

The Knudsen number plays a crucial role in understanding the behavior of gases, especially in micro and nano scale devices. It also finds application in a variety of scientific and engineering fields, including but not limited to aerodynamics, microfluidics, and chemical engineering.

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