Hydrogen gas/Pressurized storage

How much steel: Estimating the absolute bare minimum required

In physics, there is a general formula to estimate the theoretical minimum material-to-gas ratio, for any material containing any gas at high pressure.

a = 1.5 R T D / (S M)

Parameters
`a`	The ratio we're calculating Mass of the container, divided by the mass of the gas
`R`	The universal gas constant (`8.3144626 J/K/mol`) The same constant that's used in the Ideal Gas Law in physics (PV=nRT)
`T`	The maximum storage temperature Relative to absolute zero (−273°C)
`D`	The density of the material the container is made of
`S`	The tensile strength of the material the container is made of
`M`	The molar mass of the gas Which, in the case of hydrogen gas (H₂), is about `2.02 g/mol`

How this formula was obtainedHow this formula was obtained

This is based on the general formula for Spherical Pressure Vessel - Hoop Stress:

s = P r / (2 t)

's' being the tensile hoop stress on the material
'P' being the pressure of the gas
'r' being the radius of the container
't' being the thickness of the container wall

For our case, let's assume that the container is pressurized to its limit - in other words, 's' is also the tensile yield strength of the material.

We can rearrange the formula to get the wall thickness of the container:

t = P r / (2 s)

From there, we can get the volume of the container material, approximately:

Vmat = 4 pi r^2 * t
(Area of the sphere, times thickness)

Meanwhile the volume of the gas is:

Vgas = (4/3)pi r^3
(Volume of the sphere)

Therefore, the material-to-gas ratio BY VOLUME is:

Vmat / Vgas
= 3t / r
= 1.5 P/s
(Yes, everything cancels out except for 'P' and 's'. The ratio is the same no matter the size of the container.)

But we want the material-to-gas ratio BY MASS instead, so:

Mmat / Mgas = Vmat / Vgas * Dmat / Dgas
= 1.5 P/s * Dmat / Dgas

First we need to know the density of the gas (via the gas law in physics):

Dgas = P M / R T

'P' is the pressure as used earlier
'M' is the molar mass of the gas molecules (chemistry)
'R' is the universal gas constant
'T' is the temperature

From here, the final equation is:

Mmat / Mgas = 1.5 R T Dmat / (s M)
(Yes, 'P' cancels out of the equation. The ratio is the same no matter the pressure of the gas.)

Note that there is no safety factor in this equation. There is also no accounting for the materials in the gas valve etc. It calculates only the theoretical minimum material. In real life, you will need a bit more.

TALK:

Yes I know it's awkward to use 'M' for molar mass and 'Mgas' for the total mass.
Yes I know I didn't formally define 'Dmat' but whatever, hopefully people can infer.
Yes I know that relative pressure (actual gas pressure minus 1 atm) is what really matters - but that would make the formula more complicated. The simpler formula above works fine if the we assume that the gas is significantly pressurized. For low pressures close to 1 atm (i.e. a simple balloon, not what this page is about), we could use less material in theory.

Let's apply this formula to a few possible cases below:

T

tempC(60)

Maximum storage temperature

Chose 60°C because it's just above the highest weather temperature ever recorded on Earth. This doesn't account for future climate change or improper storage of the tanks. In any worst case, the tanks should be designed to vent the gas (or deform?) rather than explode, hopefully.

M

2.01588 g/mol

Molar mass of hydrogen gas (H₂)

Stainless steel

stainless_steel.density

7.80 g/cc

Density (weight per volume) of 10.9 alloy steel

stainless_steel.yield_strength

897 N/mm^2

Tensile yield strength of the stainless steel

Using "10.9 alloy steel" http://www.volksbolts.com/faq/basics.htm

(calculation loading) ^ Material-to-gas ratio

Stainless steel contains more than just iron & carbon. It also contains nickel & chromium, which are scarcer resources. Need to do an analysis regarding mineral reserves.