High-Pressure Transfill Z-Factor Compensation?

[iointerrupt]

Say hypothetically we have an empty AL80 (11.1L) and a full HC4500 (45.1L) cascade tank.

If we use the 'standard' transfill equation, we end up with both tanks at 3600psi (250bar):

equalizedPressure = (310 * 45.1) / (11.1 + 45.1) = 248 bar = 3597 psi

Obviously our little AL80 can't handle that, so we're going to need calculate it another way.

The Oxygen Hacker's Companion shows another way to calculate it, using what I'll call "transfill factor".

Simply dividing the tank water volume sizes: 45.1 / 11.1, gives a transfill factor of 4.06. If we transfill the AL80 to 3000psi (206 bar), we then subtract 206/4.06 from the HC4500:

HC4500-final-pressure = 310 bar - (206/4.06 bar) = 259 bar = 3756 psi

So that result looks pretty reasonable, the AL80 now has a full 3000psi (206 bar) in it, and the HC4500 3756psi (259 bar).

The only catch is, what about the compressibility z-factor? We took the top 750psi of the HC4500, and assumed it filled 77 cubic feet in the AL80. But the Z-Factor at 4500psi is around 1.1. So is that 750psi really more like 72 cubic feet? In which case, our final-pressure number is higher than it should be.

Seemingly we need to account for z-factor here, and if so, how? Or is there another approach to all this entirely.

 

[DiveGearExpress]

Yeah, in theory you could account for gas compressibilty using Z-factors when cascade filling although the math starts to get messy. Z-factors in SCUBA are most useful in dive planning for determining how much breathing gas volume you really have in your tanks (because divers frequently overestimate how much breathing gas they have, and a significant factor that contributes to that overestimate is compressibility.) In practice, bothering with Z-factors for gas handling cascade applications is typically not worthwhile as there are going to be some other effects at work as well... such as temperature changes and bleed/leak losses along with other "unknowns".

As you suspected in your OP, there are alternative approaches. In the compressed gas handling industry, accounting for compressibility via Z-factors is relatively uncommon. Instead the compressed gas industry relies upon Standard Cubic Foot (SCF) calculations. Standard cubic foot - Wikipedia When you see capacity tables for different gases in HC4500 and other storage bank cylinders, the volumes expressed are most often referring to SCF. If you rent a T-bottle of helium for example, the invoice will almost certainly list the volume of helium you are getting in SCF, not an 'ideal' volume calculation adjusted with a Z-factor.

In the real world of SCUBA applications I would not recommend obsessing over gas handing details like this, it can be really annoying. I once designed and operated one of the largest multiple SCUBA gas blend storage and cascade systems I've ever heard of and the math (i.e. theoretical) never ever matched up particularly well with the observations.