# How to calculate gas flow/permeation through solid metal membrane

Given the absolute pressures of air on both sides of a titanium membrane of known thickness (0.1mm) and diameter (3mm), how would one calculate the rate of gas exchange/escape through the titanium material assuming a pressure gradient? Ideal units are in cc/sec.

• Not sure how to approach this problem. But I do know the rate of diffusion across a metal membrane will also be a function of the membrane temperature. Such membranes are used as 'valves' to inject gas particles into particle accelerators. Perhaps if you research in that area. I imagine for 0.1 mm thickness the rate would be very low. – docscience Oct 13 '15 at 19:56

The Diffusion Flux of a gas through a metal membrane is theoretically given by:

$\large{J=\frac{KD(\sqrt{p_2}-\sqrt{p_1})}{\delta}}$

With:

$p_1$ and $p_2$ the pressures on the respective sides on the membrane.

$\delta$ the thickness of the metal membrane.

$K$ is a proportionality constant that connects gas concentration in the metal with pressure of the gas via:

$K^2=\frac{c^2}{p}$, where $c$ is the concentration of the gas in the metal and $p$ the pressure of the gas.

The main problem now becomes finding reliable values for $K$ and $D$. Both are likely to be very small, for example for hydrogen/iron Wikipedia lists a value of:

$D=1.66 \times 10^{-9}\:\mathrm{cm^2/s}$.

$K$ is also likely to be very small, as $c=K\sqrt{p}$ and the solubility of air in titanium metal is very small.

This means that at modest pressure differences and a membrane thickness of $\delta =1 \times 10^{-4}\:\mathrm{m}$, $J$ is likely to be negligible.

But even if the pressure was $100$ times larger on one side of the membrane than the other then J would still be only approx. $\sqrt{100}=10$ times higher.

A $0.1\:\mathrm{mm}$ thickness a titanium membrane is likely not to leak at all, at least not in 'reasonable' conditions of pressure.