Predicting Remaining Pressure in Compressed Gas Cylinder

[Saladsamurai]

TL;DR

I would like to sanity check that my tank's observed pressure is aligned with expected usage (i.e. that there is not an obvious leak, break, etc). My prediction is within 2% of measurement, but I wanted to solicit feedback if you have any.

Well, it's been nearly 10 years since my last post, and it's been about that long since I've thought about ideal gases, so here we go .

Description of Setup
I have a system that uses compressed gas cylinders as a source to slowly purge an optical payload. The source is 12x compressed nitrogen cylinders manifolded together (see this ProRack for example). There is a regulator mounted directly to the output of the 12-pack and before any gas was used, its high pressure side read 2600 psi, which is expected since this is the pressure that the tanks are delivered at. Additionally, we know that the "water volume" of each tank is 1.76 ft^3 and that each tank holds about 300 ft^3 of N2 compressed from atmosphere to 2600 psi (these numbers all agree well with Boyle's Law).

A tube runs from the output of the source regulator into a filtration system to ensure that the N2 is extremely clean before passing through the payload. This filter system has its own pressure regulator and metering valve which allows us to set the pressure and flowrate of the N2 as it exits the filtering system and enters the payload. A flowmeter downstream of the filtration system measures flowrate just before entering the payload. Upon passing through the payload, the flow exits to atmospheric pressure (14.7 psi).

The environment stays approximately constant temperature of 25 °C; the flowrate is set to 8.48 SCFH (standard ft^3/hour) at 5 psig (which again eventually vents to atmosphere).

Problem/Question
After approximately 54 hours of continuous use at approximately constant conditions (flowrate, purge pressure, temperature), the source regulator high pressure side is down from 2600 to 2250 psi. I would like to assess whether this is a reasonable drop in pressure given the consumption rate of N2.

Current Approach
My approach has been fairly simple, uses the ideal gas law and is as follows:
1) Determine initial number of moles in the source 12-pack:
n = pV/RT
--> n = (2600 psi)(12*1.76 ft^3) / [(8.3145 J/mol-K)*(298.15 K)] * [6894.8 Pa/psi) / (35.315 ft^3/m^3)]
--> n = 4324.73 moles N2 initially in 12-pack.
Note that the ratio in purple is solely for unit conversion.

2) Use the time elapsed and flowrate to calculate the volume of nitrogen expanded at atmospheric pressure:
Example: At 54 hours and 8.48 SCFH, we have expanded a volume of
V_e = 54*8.48
--> V_e = 458 ft^3 of N2 expanded.

3) Determine how many moles of N2 were removed from the tank during this expansion:
pV = nRT -->
n = (14.7 psi)(458 ft^3) / [(8.3145 J/mol-K)*(298.15 K)] * [6894.8 Pa/psi] / [35.315 ft^3/m^3
--> n = 530.24 moles N2 removed from source during 54 hour expansion.

4) Therefore the number of moles that remain in the tank is:
n = 4324.73 - 530.24
--> n = 3794.5 moles N2 remain at 54 hours.

5) Using ideal gas law to calculate the pressure of these moles is straightforward:
p = nRT/V
--> p = (3794.5 mol)*(8.3145 J/mol-K)*(298.15 K) / [(12*1.76 ft^3)*(6894.8 Pa/psi)*(35.315 ft^3/m^3)]
--> p = 2281 psi at 54 hours.

Discussion:
The predicted and observed values appear to be in good agreement from my perspective (within 2%). I am curious if this really is a linear problem though? My approach was that the current pressure inside of the tank never depended on a previous value of pressure other than the initial value. I think this makes sense since we are at a point in the life of the supply where there is more than enough source pressure and volume to supply my relatively low flowrate and delivery pressure. I presume that if/when the source pressure starts getting considerably closer in magnitude to my delivery pressure, things start to get funny ... and hopefully I've sized my supply to never get close.

I am open to feedback/discussion on this topic. Any issues you see with this approach?

Thanks for reading.

 

[hutchphd]

The question is how good is good enough?
In my head I checked your data (to ~±10%) and got a use rate of 8 moles per hour so 430 moles. So your arithmetic looks error free. How good is your flow meter? How good is your temperature control?
Probably the worst error is ideal gas assumption for N₂at that pressure (you can and should look this up)but this should get better with pressure reduction
So measure the system into lower source pressure if possible and continue the comparison. Make a graph of calculated vs actual. Put it on the wall and look at it whenever you need reassurance !

 

[jrmichler]

TL;DR Summary: I would like to sanity check that my tank's observed pressure is aligned with expected usage (i.e. that there is not an obvious leak, break, etc). My prediction is within 2% of measurement, but I wanted to solicit feedback if you have any.

The predicted and observed values appear to be in good agreement from my perspective (within 2%).

If you are measuring pressure with standard pressure gauges, be advised of their accuracy specifications. For example, typical Ashcroft (randomly chosen well known manufacturer) pressure gauges have accuracy specification of +/- 3%: https://www.ashcroft.com/products/pressure/pressure-gauges/1005-1005p-1005spressure-gauge/. And that tolerance applies to brand new gauges.

and 8.48 SCFH

Check the accuracy specification on your flowmeter. I'm sure that it is much worse that +/-0.01 SCFH.

approximately constant temperature of 25 °C

Few thermometers are accurate to better than +/-1 deg C. Even the digital ones that read out to 0.1 deg C.

Plus everything that @hutchphd said.

TL;DR Summary: I would like to sanity check that my tank's observed pressure is aligned with expected usage (i.e. that there is not an obvious leak, break, etc).

And a big thumb up that you are cross checking.

 

[Dullard]

I run into this problem occasionally - more often with dewars (in my case), but:

It is useful to think of the initial capacity of your bottle array in Bar-Liters and your consumption rate in Bar-Liters/Hr.
you have 106K Bar-L and 240 Bar-L/Hr, respectively (I noted that 8.48 SCFH is suspiciously close to 4.00 SLPM).
you have about 440 Hrs of capacity (to empty).

The volume of your tank array is 600 L. The expected pressure at any time may be found by subtracting the capacity used from the initial capacity and dividing by the volume.

This isn't fundamentally different from your approach, but is a lot easier to do in your head.