Calculating refills from a higher pressure tank

[WetSEAL]

EDIT: the question has been solved. The confusion below arises from the fact that it's not actually a 97 cf tank, it's a 550 cubic inch tank (that could hold up to 97 cubic feet of uncompressed air)

Suppose I have a 97 cf tank at 4500 psi, and a 13 ci tank at 3000 psi. Every time the little tank gets down to 2000 psi, I refill it from the big tank. How many times can I refill it to 3000 psi?

My attempted solution:

For the small tank, 13 ci * (1 cf/1728 ci) = 0.00752315 cf

On the small tank, using the ideal gas law, I first compute the volume of air if reduced to 2000 psi:
V3 = 3000/2000*13 = 19.5

This means that if I run it down to 2000 psi, then I have 13/19.5 or 66% air left...which means I used 33% of the original air. Therefore, each time I refill, I am adding back 0.33*0.00752315 cf = 0.0024826395 cf at 3000 psi

For the big tank, using the ideal gas law, I compute the equivalent storage capacity at 3000 psi.
V2 = 97*4500/3000 = 145.5 cf at 3000 psi

However, I wan't to ignore the air once it gets down to 3000 psi, so I have to calculate the volume of air left at 3000 psi:
97/145.5 = 66%

So, my "real" volume of air (before getting down to 3000 psi) is actually 0.33*145.5 = 48

So, I can do 48 / 0.0024826395 = 19,334 refills.

This seems insanely high, so I'm assuming I made a math or logic error...can anyone sanity check me here?

 

[DiveClimbRide]

*It appears I don't understand imperial conversions. Please disregard everything below.*

For the big tank, using the ideal gas law, I compute the equivalent storage capacity at 3000 psi.
V2 = 97*4500/3000 = 145.5 cf at 3000 psi

Looks like you have a problem here. How do you have more air at a lower pressure? You should be doing V2 = 97*3000/4500. Which gives you aprox 64cf so the useable air is 33cf. Also is your smaller tank capacity in cubic inches? A tiny tank will get a lot of refills from such a large tank.

 

[johndiver999]

You need to define the capacity and fill pressure of the 97 cu ft tank. Is it 97 cu-ft at 3000 psi? or 97 cu-ft at 4500 pis?

Lets make it simple and assume you have 100 cu-ft at 3000 psi in big tank.
If you fill it to 4500 pis you will add 50% more air so that is 50 cu-ft (in my example)

So the available "extra" volumetric capacity above 3000 psi is roughly 50 cu-ft

If each pony fill is 1/3 times 19 = about 6.3 cu-ft

so 50 cu-ft / 6.3 cu-ft is around 12.6 times.. Not sure about the very last fill, but that seems like a rough way to do it.

 

[WetSEAL]

Looks like you have a problem here. How do you have more air at a lower pressure? You should be doing V2 = 97*3000/4500. Which gives you aprox 64cf so the useable air is 33cf. Also is your smaller tank capacity in cubic inches? A tiny tank will get a lot of refills from such a large tank.

It's volume. the same air takes up a greater volume when you reduce the pressure from 4500 to 3000 psi.

P1*V1 = P2*V2
4500*97 = 3000*V2
V2 = 97*3000/4500 = 145.5 cf

 

[WetSEAL]

Lets make it simple and assume you have 100 cu-ft at 3000 psi in big tank.
If you fill it to 4500 pis you will add 50% more air so that is 50 cu-ft (in my example)

So the available "extra" volumetric capacity above 3000 psi is roughly 50 cu-ft

Okay, that's the same as what I did above. My exact calculation above was 0.33*145.5 = 48 cu-ft, basically the same as your approx

If each pony fill is 1/3 times 19 = about 6.3 cu-ft

Looks like you got the units mixed up here. The pony is 13 cubic-inches. Also not sure why you took 1/3 of 19.
I calculated 19.5 as the volume of space that would be taken up if you reduced the pressure from 3000 psi to 2000 psi.
When the pony needs a refill, it's at 2000 psi in 13 cubic inches, which means that you have 13/19.5 = 66% air molecules remaining. This means that 1/3 of the original 13 cubic-inches was used up. 1/3 of 13 cubic inches equals 0.0024826395 cubic feet.

so 50 cu-ft / 6.3 cu-ft is around 12.6 times..

Okay, you're doing the same final division that I did, except that it should be 0.0024826395 cubic feet instead of 6.3 cubic feet when you take into account the two corrections above..

 

[Brett Hatch]

I haven't double-checked either of your math, just want to point something out here about the units. The conversion factor is 1 cubic foot to 1728 cubic inches. Multiplying @johndiver999 's result by that conversion yields about about 21 thousand fills, which is pretty consistent with the 19 thousand you found (since you're rounding off slightly differently).

I think the "confusing" part of this is that a 13 cubic inch tank is really, really, really small. 13 cubic inches of air compressed to 3000 psi is about 1 ml, or 1/5th of a teaspoon. So yeah, just to sanity check this, we can flip the question around: how many 1/5th of a teaspoons can you take out of a scuba tank? Of course the answer is gonna be a huge number

Edit to add: it sounds a bit like this is a contrived question that might not really reflect what you are actually trying to figure out. What is the original problem that you are trying to solve? Does it really involve a 13 cubic inch tank?

 

[WetSEAL]

I haven't double-checked either of your math, just want to point something out here about the units. The conversion factor is 1 cubic foot to 1728 cubic inches. Multiplying @johndiver999 's result by that conversion yields about about 21 thousand fills, which is pretty consistent with the 19 thousand you found (since you're rounding off slightly differently).

I think the "confusing" part of this is that a 13 cubic inch tank is really, really, really small. 13 cubic inches of air compressed to 3000 psi is about 1 ml, or 1/5th of a teaspoon. So yeah, just to sanity check this, we can flip the question around: how many 1/5th of a teaspoons can you take out of a scuba tank? Of course the answer is gonna be a huge number

Edit to add: it sounds a bit like this is a contrived question that might not really reflect what you are actually trying to figure out. What is the original problem that you are trying to solve? Does it really involve a 13 cubic inch tank?

The specific question comes from paintball, which is the reason for the smaller 13 cubic inch tank, but I figured that the math behind it relates equally well to scuba

I think I discovered the source of my confusion. The manufacturer of the 97 cf tank says "97 cubic ft (or 550 cubic inches)": (Air Tanks For Sale Discounts for PCP (Pre-Compressed-Pneumatic) Air Gun by AirTanksForSale )

I was doing my calculations in terms of cubic ft so didn't notice this, but I think 97 cubic ft is NOT equal to 550 cubic inches! Using 1728 cubic inch per cubic ft, it would be 167,616 cubic inches. This would explain why I was getting such ridiculous numbers of refills. I believe this tank is actually only 550 cubic inches...not sure how the manufacturer is doing their conversion?

Secondly, I realized that by going from 4500 psi down to 3000 psi on the main tank means that you only have 1/3 usable capacity, and also ironically if you refill the small tank at 2000 psi down from 3000 psi, that means you used 1/3 of it. This means that the full equation can be simplified and the 1/3 in numer and denom actually cancel out,

(550 ci/3) / (13 ci/3) = 550 ci/13 ci = 42

So, in other words, if I start with a 550 cubic inch tank at 4500 psi, and use it to refill a 13 cubic inch 3000 psi tank every time it gets down to 2000 psi, then I believe the answer is 42 refills...which makes sense because this is also the meaning of life. I believe that my original calculation was correct, as I stated it...I was just using a ridiculously sized tank

 

[jborg]

550 cubic inches is 9 litres which is a sane tank size, assuming we’re talking about water volume. So I think it’s just the manufacturer being confusing in their “97 cuft (of compressed air) or 550 cubic inches (of water volume)” spec.

Once you’re dealing with water volumes and pressures on both sides you’re essentially using the metric way of doing tank sizes, just with different units, and the maths actually become a bit simpler.

 

[rhwestfall]

search here on SB for a discussion of transfilling... there is a discussion referencing a few pieces of software...

 

[DiveClimbRide]

confusion below arises from the fact that it's not actually a 97 cf tank, it's a 550 cubic inch tank (that could hold up to 97 cubic feet of compressed air)

FIFY