Paging Pneumagger and other types good at Math Fills off a 321 CF tank?

[matteusz]

OK so I have the option to buy a 321 CF 3500 PSi tank for 120$ and it will cost 60$ to have it filled. It would cost me 80$ for the fill adapter.

Or I can buy three 80 cf scuba 3000 psi tanks for 450$ and it will cost 25$ to get them filled. Fill adapter would be 3 of the yoke types for 150$ roughly.

Which one will give me cheaper fills per use with my smaller tanks. So how much will it cost me to fill my smaller tanks from each one and is it worth the higher fill price or the higher purchase price?

How many fills would I get off the 321 CF tank? I think I have a pretty good idea of getting some 20 fills above 2000psi off of three 80 cf tanks. Is that correct?

If someone does some impressive math on this (or comes up with a great and workable alternative) I will give them a discount on a Logic frame or some free AGD stickers in the mail to make it worth your while.

Mat

[Engus]

the value of each setup kinda depends on how often you plan on using them. and where to do you get that ripped of for air fills? I can get a scuba filled for $5

[matteusz]

the value of each setup kinda depends on how often you plan on using them. and where to do you get that ripped of for air fills? I can get a scuba filled for $5


Um no points for you. I want to know the COST of each fill as in dollars and cents per fill on my paintball tank. And that is the going rate at the local scuba shops.

[Pneumagger]

Well, there's 2 ways to approach the problem... a fill decrement analysis and raw work approach. I'll do the raw work approach first in generic form and then answer you query with realistic bounds. Next I'll do the decrement approach (probably the answer your familiar with). The raw work solution will tell you the most economical solution via which has the most air per dollar and the decremental analysis will tell you exactly how many fills between bulk recharge you will get.

[Pneumagger]

Compressed air is a pressure (pounds force per square inch) contained in a volume (cubic inches). If you multiply a pressure vessel's pressure by its volume you get a number with the units of "pounds force by inches" or lbf*in. The funny thing here is that a force multiplied by a distance yields a unit of work (lbf*in for example) - so we have effectively calculated the amount of work a tank of compressed air can do. The amount of lbf*in a tank is capable of containing is directly proportional to the number of fills it can dish out. Now all one must do is determine the total work per dollar spent to arrive at the most economical solution.

(max pressure of bulk tank) x (Hydrostatic Volume)
---------------------------------------------------------------- = Value (lbf*in/$)
(Cost per fill)

This solution simply yields overall work per dollar in the tanks from max fill to empty and are not yet applicable to your requests. We must recall that you wish to maintain 2000psi as the minimum utility limit. To factor in that limitation we must cut off all work being done under the 2000psi threshold by simply subtracting 2000psi from each tank's max pressure. We will call this value "delta pressure of bulk tank". It's important to note the 321CF tank has a 1.7:1 capacity advantage per fill. However, you will see that the smaller tank's cheaper refill rate dominates the larger tank's capacity advantage.


(delta pressure of bulk tank) x (Hydrostatic Volume)
---------------------------------------------------------------- = Value (lbf*in/$)
(Cost per fill)


(1500 psi) x (2329 cu.in.)
--------------------------------- = 58226 lbf*in/$ <--- One 321CF tank
($60)


(1000 psi) x (2055 cu.in.)
--------------------------------- = 82200 lbf*in/$ <--- Three 80CF tanks
($25)

We can clearly see the 1.42:1 fill cost efficiency ratio favors the three 80CF tanks.
Now we will factor in the cost of tanks and fittings. This is where the smaller cheaper to fill tanks really shine in performance.

(tanks) + (fittings) + (cost per fill)(relative capacity advantage)(total number of fills to date) = Total expenditure

For the 321CF tank:
($120) + ($80) + ($60)(X) = Total Expenditure

For the three 80CF tanks:
($450) + ($150) + ($25)(1.7)*(X) = Total Expenditure

So which is more important: tank capacity or economy of refill? Now we equate these 2 equations to determine the switch-over point. Here is what we can determine from comparing the two alternatives (in english):


Right off the start the 321CF tank delivers cheaper overall air. It is a cheap setup high refill cost option, the more you get it refilled the less economical it gets. Conversely, three 80CF tanks are expensive to setup but cheap to fill. This means as the number of bulk refills approaches infinity you save 30% in costs in choosing the three 80CF tanks at the penalty of refilling 1.7 times as often. This is because three 80CF tanks have a 1.42:1 fill cost efficiency advantage but your filling tanks that only have 59% as much work in them. Basically, long term operation economically favors three 80CF tanks and short term operation favors the 321CF tank. So what actually determines "long term" and "short term"?

If you purchase the large 321CF tank, you must not refill the tank more than 22.86 times or else you were better off buying the three 80CF tanks. Likewise if you purchase the three 80CF tanks instead, you must refill them at least 38.9 times or else you were better off purchasing the 321CF tank. This is your end solution.

Additionally, the best solution you will find (short of compressing your own air) is to get the bigger single tank and have it filled somewhere using the little tanks' fill value. Although, even this proposed solution's advantage diminishes asymptotically as the number of refills increases ad infinitum.



[disclaimer: All of these solutions are based off general mfg specifications, do not account for any losses and phase transformations, and will not be 100% precise - albeit close. These conclusions assume equal performance delivered from both systems and both systems refilled to capacity. I could not find mfg specifications on a 321cuft 3500psi tank. Damn that's a big tank. Anyways, I simply ran a linear extrapolation using generic capacity --> compressed volume calculations. It should be sufficiently accurate for this problem.]

[Pneumagger]

(I actually took a previous post I had mede on the subject and modified it for use for your conditions)

PV=nRT
Assume nRT = constant
Assume V(s) = internal volume of scuba
Assume P(1) = Pressure in scuba before fill
Assume P(2) = Pressure in scuba and pb tank after fill
Assume V(t) = internal volume of paintball tank
State 1 = value before fill
State 2 = Value after fill

-----------------------

You can see the tank pressure in the bulk cylinder will drop with each tank fill. You decided on a minimum utility level of 2000psi before filling bulk tanks and the assumed tanks we will be filling are 68ci tanks. An 80CF tank carries has roughly 685ci of hydrostatic volume and carries 3000psi, three of them have 2055ci of volume. A 321CF tank would have somewhere on the order of 2329ci and carries 3500psi. With the above assumptions, P(1)*V(s) = P(2)*V(2) quickly simplifies to for your case:

P(1)*V(s) = P(2)*(V(s)+V(t))

Tabulated out, assuming a 3000psi filled 80CF cuba filling an empty 68/3000 tank, while taking dropping scuba pressure into account we see that...


Three 80CF tanks @ 3000psi
Fills....Scuba Pressure...Tank Pressure
0.......3000.000000.......0
1.......2903.909562.......2903.909562
2.......2810.896915.......2810.896915
3.......2720.863476.......2720.863476
4.......2633.713822.......2633.713822
5.......2549.355583.......2549.355583
6.......2467.699352.......2467.699352
7.......2388.658581.......2388.658581
8.......2312.149498.......2312.149498
9.......2238.091012.......2238.091012
10......2166.40463........2166.40463
11......2097.014373......2097.014373
12......2029.846697......2029.846697
13......1964.830411......1964.830411



One 321CF tank @ 3500psi.
Fills....Scuba Pressure...Tank Pressure
0.......3500.00000......0
1.......3400.70922......3400.70922
2.......3291.784007.....3291.784007
3.......3186.347685.....3186.347685
4.......3084.288503.....3084.288503
5.......2985.498292.....2985.498292
6.......2889.872346.....2889.872346
7.......2797.309312.....2797.309312
8.......2707.711087.....2707.711087
9.......2620.982705.....2620.982705
10......2537.032246.....2537.032246
11......2455.770733.....2455.770733
12......2377.112038.....2377.112038
13......2300.972792.....2300.972792
14......2227.272298.....2227.272298
15......2155.932441.....2155.932441
16......2086.87761.......2086.87761
17......2020.034615.....2020.034615
18......1955.332612.....1955.332612

Factoring in quick or frequent filling (allowing the bulk tank to chill) which is more representative of an adiabatic situation, the fill efficiency will worsen. Also, this chart is indicative of filling an empty 68ci tank, obviously topping off a 68ci uses less air. I prefer the first method of solving the problem as it does not need to take into account various fill techniques or even what kinds of paintball tanks your filling - it's a much more elegant solution.
The three 80CF cascade station will give about 12 fills over 2000psi on empty 68ci tanks. The 321CF station will give you about 17 fills over 2000psi on empty 68ci tanks.
Obviously, if your just topping off 68ci tanks you'll pick up a substantial number of fills on top of that figure.

Looking at the economy of the situation:
(tanks) + (fittings) + (cost per bulk refill)(bulk refill per pb tank)(X) = Total expenditure

For the 321CF tank:
($120) + ($80) + ($60)(1/17)(X) = Total Expenditures

For the three 80CF tanks:
($450) + ($150) + ($25)(1/12)(X) = Total Expenditure

Setting these two equations equal to each other lets us know the break-even point for this solution is 276 paintball tank fills. If you plan on filling less than 276 68ci tanks for the life of your fill station, buy the big 321CF tank. If you plan on filling more than 276 68ci tanks, buy the three 80CF tanks.

(note that based on the decremental fill chart the Raw Work solution suggests the tanks should break even around 400 tank fills.)
-----------------------------------

Looking at the original Raw Work solution and the latter Decremental solution it is apparent that latter solution says your fill efficiency will be substantially worse than the Raw Work equations indicate. This may be puzzling at first but make perfect sense when you look at the basic assumptions used in setting the equations up. The decremental solution assumes you are filling EMPTY 68ci tanks - for example prior to every refill you would vent out your 68ci tanks. This is not very realistic for a majority of the fills encountered as people generally top off at 1000psi. The Raw Work solution on the other hand is assuming efficient air usage of the pressures in the bulk tanks over 2000 psi. In real life when you top off, you use less air and get more partial fills. Obviously, the more you top off and the more efficient you are, the closer your break even point will move from 276 fills to ~400 fills.

CONCLUSION: Your break even point filling 68ci tanks lies somewhere between 276 fills and a theoretical maximum of ~400 fills. It all depends on how you top your tanks off and how efficient you are with your fills. Let's say your somewhere right between those two figures and your break even point (based off the combined solutions) is an estimated 340 individual tank fills. Basically, if you plan on filling less tanks than that for the lifetime of your fill station you should purchase the 321CF tank. Likewise, if you plan on filling more tanks than that you should purchase the three 80CF tanks.

I hope this solution helps somewhat.
:cheers:

[matteusz]

Ah now that is some beautiful work. You have mail. :shooting: