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Transition flow

There is a gradual transition from molecular to laminar flow which can be interpreted as both flows are present simultaneously or: at the entrance of the leak the flow is laminar which gradually changes, till at the end of the leak it is molecular flow.

The path of one molecule in transition flow. It shows collitions with the wall and with other molecules.

The mathematical description of this condition is difficult. There are some formulae available, all of them have some restrictions. The most simple formula is from Burrow. He combined the formulae for laminar and molecular flow.

$q = \frac{\pi \cdot r^4}{16 \cdot \eta \cdot \textup l}\left({p_1}^2-{p_2}^2 \right) + \frac{\sqrt{2 \pi}}{6} \; \sqrt{\frac{\textup R \cdot \textup T}{M}} \frac{d^3}{l} \left(p_1 - p_2 \right)$

$q = \textup{leak rate} = Pa \cdot m^3/s$

$\eta = \textup{dynamical viscosity of the gas} = Pa \cdot s$

$p_1 = \textup{the higher pressure} = Pa$

$p_2 = \textup{the lower pressure} = Pa$

$R = \textup{universal gasconstant} = 8,314 J/Mol \cdot K$

$\small \textup{where 1 Joule} = 1 Nm = 1 Ws$

$T = \textup{absolute temperature} = K$

$M = \textup{relative molecular mass} = He = 4$

$d = \textup{diameter of the leak} = m$

$l = \textup{length of the leak} = m$

This formula can be used for rough calculations, if one takes the geometrical dimensions of an idealized leak capillary with round cross section and length much longer than diameter. One has to estimate, if molecular or laminar flow predominates and use the formula of Poisseuille or Knudsen, to calculate the dimension of the idealized leak.