Adequate Cascade System

[rx7diver]

Today I chatted with a university scuba student about the current "Cascade Storage Banks" VDH thread. She asked how someone would design a cascade system "adequate for her needs." I told her that once she made certain simple assumptions about capacity, usage, etc., then doing so is straightforward, but she absolutely would need to know how to compute the tank pressures resulting after a scuba cylinder is equalized with a cascade cylinder.

I stepped her through a quick equalization example, and then, to see if she really understands, I challenged her to spec out an "adequate" cascade system using some realistic conditions/assumptions I gave her. No internet "help" allowed! Excel spreadsheet calculations permissible, though she must explain them to me.

She took the bait!

If there is interest here, I will post her solution when it's available.

rx7diver

[tbone1004]

that's something I did back in college too! It's pretty straight forward if you assume the compressor is not running, but it gets real sporty in the formula realm if you turn a compressor on...

[rx7diver]

Okay. Here are the assumptions I asked her to use:

1. You and your dive buddy will dive every other weekend, two single-tank dives each weekend day each of you, year-round. (So, eight scuba cylinders worth of diving, total, every two weeks.)

2. You will be diving "semi-local." You two will drive out Friday afternoon/evening or Saturday morning, dive Saturday and Sunday, and return Sunday afternoon/evening. You each have four scuba cylinders.

3. Your home fill station, a cascade system, will be used to fill both your and your buddy's scuba cylinders.

4. You would prefer to run your compressor only once every four weeks. (So, you would prefer a fill station that has enough capacity to fill 16 scuba cylinders on a single charge, 13 times per year.)

5. You and your buddy are both diving either steel 72's (71.2 cu ft at 2,475 psig = 2,250 psig + 10%, filled to a max pressure of 2,250 psig), or Luxfer Al 63's (63 cu ft at 3,000 psig), or Luxfer Al 80's (77.4 cu ft at 3,000 psig). CHOOSE ONE TYPE OF CYLINDER. Assume your scuba cylinders gauge 200 psig when they need to be filled.

6. Each of your cascade cylinders has a capacity of 444 cu ft @ 4,500 psig. You will fill the cascade cylinders to a max pressure of 2,500 if you chose the 72's, and you will fill the cascade cylinders to a max pressure of 3,300 psig if you chose either of the 3,000 psig aluminum cylinders.

Okay. Spec out your cascade system. Explain your solution. Show your work. No internet "help" allowed. Excel and/or programming is permitted, but you must explain to me what you've done.

Good luck.

rx7diver

P.S. @tbone1004, she is not a math or engineering or physics/chemistry student, so we won't assume the compressor is running. No calculus nor differential equations is necessary here. And we'll assume Ideal Gas Laws work here.

[ScubaLawyer]

You guys are just making my head hurt now.

[rx7diver]

I'm making my own head hurt, @ScubLawyer! Now I've got to solve this to check her answer!

This week is finals week, though, so I'm thinkin' she won't come back with her solution until next fall when school resumes!

rx7diver

[tbone1004]

that's rude with the fill pressure of the cascade.... Also super annoying with our indirect measurement system, sooooo much easier with direct tank measurements

Did you give her one whip or two? If one whip, that is a lot of iterations and you should probably use something more substantial than excel and having a background in programming is certainly worthwhile.

Now, the short way around this.
Some assumptions.
There is no volume in the hose-it's not insignificant if the hose is pretty long, but you have to assume it is not there
Have to assume that temperature is constant-the math to factor that in is not feasible in excel and would require a lot more robust software to model properly. We know that the tanks will be warm, but you only gave a 300psi buffer in the banks which is not enough to counter the heat and assume a full fill.
Assume a single fill whip-which would suck for 16 bottles, but oh well.

Assuming AL80's, 200psi reserve and 77.4cf when full, means 72.25cf needed per bottle. 16 bottles yields 1156 cf total.
Cascade bottles at 3300psi minus 200psi yields 306cf available per bottle. Bare minimum of 4 bottles just to meet the cubic footage requirement.

With 3300psi, you really made this quite excessive. The first bottle will equalize to ~2640psi, and the 16th bottle will equalize at ~260psi assuming no losses. If you up the pressure to 3750 then you can at least get the first bottle full out of the first tank and get ~275psi in the last one.

I'm not going to write a sheet to do all of this, but it suffices to say that the limited bank pressure is going to ultimately cause you to need about as many cascade bottles as you do scuba tanks in order to get them all full. I'd SWAG it around 12 banks at 3300 to fill 16 80's to 3000psi with near guaranteed carpal tunnel at the end of a weekends worth of cascading unless you have ball valves on everything.

[rx7diver]

I had a bit of free time earlier this morning and, so, ran through this exercise because, while it seems straightforward, I had actually never solved it before, and I was curious. (I kinda "knew" what to expect, though, because of previous experiences IRL. More about this later.)

To summarize, we sought to configure a cascade system that can fill 16 single scuba cylinders without recharge, and without the compressor running, and assuming:

1. The scuba cylinders are Steel 72's (71.2 cu ft @ 2,475 psig = 2,250 psig + 10%), all starting off at 200 psig, but each filled only to 2,250 psig.
2. The cascade cylinders are 444 cu ft @ 4,500 psig, but all starting off at max pressure 2,475 psig.
3. Scuba cylinders are filled one at a time very slowly.

My solution uses only Boyle's Law (and relative proportions), and nothing more. I used an Excel spreadsheet: scuba cylinders are listed left to right, and cascade bottles are listed top to bottom. Results for the final scuba tank (the 16th scuba tank are) are shown below.

P1 and V1 correspond to the 16th cylinder.
P2 and V2 correspond to the various cascade cylinders.
P3 reflects the pressure (psig) remaining in the cascade cylinder after "equalizing."

The upshot: My solution requires eight {8} cascade cylinders--although the final cylinder remains almost full (since 2,464.7 psig remain of its starting 2,475 psig).

Caveat: I ran through this quickly. I'm reasonably confident my calculations are correct, but...

ScreenShot_20190516.jpg

rx7diver

ETA: Not sure why my png screenshot isn't showing up when I drag and drop it into the message box. I'm out of time now, so I'll try to resolve this later.

[rx7diver]

"The rest of the story."

I alluded above to my previous experiences "in real life" (IRL). Here are the details:

I took my university's scuba course in 1986 when I was completing my terminal graduate degree, and became a TA for the course for several semesters until I graduated. I was one of the few TA's the instructor/professor allowed to man the fill station (including run the compressor). The course used Steel 72's exclusively at that time. At the time I took the course, the instructor already had been teaching this course that he, himself, designed, for many years.

Early in the instructor's tenure, there had been a catastrophic scuba compressor accident at a public university in an adjacent state. IIRC, someone--a university student, I believe--was either killed or maimed by an explosion of some sort. Because of this accident, my instructor was extremely safety cautious.

My instructor set up our fill station to consist of a government surplus Worthington compressor (intake air plumbed through the roof of the natatorium; manually-operated water separators after each stage; constant-flow, freshwater-jacketed lines between stages; a single, large filter stack that he packed manually, himself; over-pressure relief valve; etc.) connected to *six* cascade cylinders, themselves connected in series via pigtails. A single fill whip was connected to the first cascade cylinder. Scuba cylinders were filled in a water-filled garbage can.

I don't recall the exact characteristics of these cascade cylinders, but I remember they had a 3,500 psig service pressure.

Our rules were to fill the cascade system to a max pressure of 2,475 psig (2,500 psig, actually), and to fill the scuba cylinders to a max fill pressure of 2,250 psig--even though the instructor had all the scuba cylinders and the cascade bottles hydro-tested every *three* years, with the scuba cylinders tested for a "+" rating.

Each of our wet sessions had about 12 scuba students. (Six pool lanes, a buddy pair each lane.) So, potentially six "empty" tanks after each hour-long wet session. The fill station I describe above had sufficient capacity to fill all 12 scuba cylinders without recharge.

(Also, the compressor had high enough SCFM to top off the cascade system in less than an hour--so that the scuba cylinders could be refilled and used the next hour, by the next class. Typically, there were three hour-long wet sessions that met twice a week, with "optional" Sunday wet sessions.)

Clearly, the instructor had built layers of safety into his set-up: It was almost impossible to seriously overfill a scuba cylinder using his setup! (The OPV was set just north of 2,500 psig, IIRC.)

So, I kinda "knew" in advance that the above exercise would end with only a relatively few cascade cylinders being needed. Indeed, my intermediate calculations show that only six cascade cylinders (rather than eight) are needed if we had specified 12 Steel 72's (rather than 16) in the above exercise--consistent with my experience IRL.

Very cool!

rx7diver