Transient Tank Fill Rate

[Jieve]

I have an application lab testing high pressure pressure sensors using an argon filled welding cylinder as the pressure source. Solenoid valves open and close to feed and bleed into a fixed volume where the pressure is measured, and I need to size the solenoid valves from a selection of available valves, given their Cv values. Ideally this would be dynamic testing, so I am trying to determine fill time of the test volume.

My thermo in this area is a little (extremely) rusty, but I found equations for the Cv / volumetric flow rate conversions of gases here (although not sure how they were derived):

https://www.idealvalve.com/pdf/Flow-Calculation-for-Gases.pdf.

So my first thought was of trying to solve this numerically as a sort of quasi-static problem, and I could use some guidance to know how to go about this.

What I do know: I know the starting T, P, and V of the cylinder, so can calculate the mass of gas in cylinder from the ideal gas law. Similarly, I know the starting conditions of the air in the test volume, so can calculate the mass there (of air). Although as a starting point, I would probably calculate assuming Argon in the test volume instead of air, to simplify the problem.

The rough idea in my head: Using the test volume as the control volume, If I were to do this numerically, and assume a small increment of dt, then knowing density I could get the mass flow from the equation in the above link, and calculate the total mass in the test volume after time t0 + dt. From there, I’d want to be able to somehow calculate pressure at each time until the time at which the pressures are approximately equal, and see at which time point this occurs.

However, first of all, I’m not sure what value to use for density, since the density of gas in the cylinder is much higher than that of the test volume (and is a function of time). Also, even if I have mass, I have V, R, and m, but don’t have T, so I’m not sure how I’d calculate P from the ideal gas law.

Another way I thought about going about this is as a first law problem. I’ve simplified the transient filling process, assuming adiabatic due to high fill speed, to dU = mdot, in(t)*hin(t) dt, but not sure where to go from here. mdot,in(t) could presumably be calculated at each step if density were known from the previously mentioned equations, but h(t) would be a function of T only for an ideal gas, and I don’t have this value.

I’d really appreciate if someone could provide some direction. Thanks for any input.

 

[LittleInch]

Your first issue I think, though it is difficult without numbers like volumes pressures etc, is the initially you will be operating in the choked flow area which is more complex until your absolute pressure ratio is about 1.8.

Why don't you just use a pressure regulator?

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[1503-44]

The gas is reducing its pressure from that in the cylinder.
Some cooling will occur.

The problem of what pressures and temperatures and pressure will be reached when the test cylinder is full can be solved by
P1 V1 / T1 = P2 V2 / T2 and
P V = n R T

P1, P2, V1, V2, T1 are known. Solve for T2

Time does not enter into the picture until you introduce a flow rate. Divide (V2-V1)/Δt and you will know the flow rate. At constant flow rate, its no problem.

If the flow rate will be determined by the pressure difference between weld cylinder and test cylinder, The dynamics can be simulated. Assume a constant flow rate over a time step. Calculate P2 at t2 from the flow rate change in volume. Recalculate flow rate from the resulting P1-P2. Adjust flow for being choaked or not. Recalculate ΔV and P2 as necessary. Do that iteration until any differences in flow rate and pressure is resolved satisfactorly (i.e. the errors are acceptable). Go to the next time step and do the same. At reaching the last time step, P2 must equal the final value as predicted from the initial ideal equations. If it does not, adjust your flow rate in the first time step and recalculate over all time steps again. You can use non constant time steps, making them smaller to improve accuracy, such as when initial pressure drops are high, or longer when pressure drops are less, as you like.

--Einstein gave the same test to students every year. When asked why he would do something like that, "Because the answers had changed."

 

[LittleInch]

A lot depends on the actual numbers.

So if e.g. the mass in the little cylinder / small volume test thing is less than 10% of the mass in the big cylinder, you can probably assume a constant pressure in the big cylinder to make life easier.

Ditto what the pressure range is. Welding cyclinder usually start at about 220bar?. What's high pressure in your test volume?

If it's less than say 50 bar your mass flowrate won't change as you're in choked flow and the upstream pressure is what determines your mass flow rate so it is constant up to about 100 bar or more.

Then it just becomes a transient pressure difference / flow.

Why not just use a pressure regulator?



Remember - More details = better answers
Also: If you get a response it's polite to respond to it.

 

[Snickster]

It is not clear what you are trying to accomplish based on your first post. Could your state your goal very clearly and how you intended to go about it. It appears you are trying to determine the filling time of the test volume base on a calculated time dependent mass flowrate of gas from the cylinder. If so this is a very complex problem and at best any very theoretical set of equations like that will just give you an approximation. Just use a regulator as suggested previously.

 

[Jieve]

Thank you all for the replies.

@Snickster: we use pressure sensors in a process application, and we need a somewhat quick and dirty way to dynamically test them in the lab before deploying them. The range is 2000-3000psi. Argon welding cylinders (125ft3) aren't expensive and are generally charged to about 2500psi, so the idea is to be able to thread the sensor into the end and use solenoid valves to quickly (ideally at 10Hz but could be slower) cyclically fill and evacuate a small test volume to ensure the sensor is good. I'm purchasing off the shelf solenoid valves for this and have their Cv values, so need to determine the fill times for different valves / test volumes to see if this is realistic or if I need go a different route.

I stated this problem in terms of unknown volumes, because I'd like to make sure I understand the thermo theory / calcs should I later have another similar problem to this, in case there are significant simplifying assumptions if I were state that the test volume will likely be very small relative to the cylinder.

@LittleInch: The value of the exact test pressure isn't too critical (could be 1200-2500 psi), the main thing is that we see the sensor output produce full range (0 - cylinder psi) dynamic readings. If the cylinder starts getting low below what we determine is a good test pressure then we'll just have it recharged.

@1503-44: Thank you for that. I'll work through this and report back as I have questions, which I think I most certainly will.

 

[Compositepro]

Pressure calibrators are an off-the-shelf item. Why reinvent the wheel?.

 

[georgeverghese]

This is not an ideal gas step change. Isenthalpic dt during pressure drop occurs because the gas in not ideal. And clearly not for argon at 2500psig start pressure, since z is much less than 1.0. In an ideal gas, there is no dt for isenthalpic letdown.
From my experience, unless you are willing to write out a complicated program for calculating real gas z, you wont get any decent results without a simulator for getting sizing case Cv values for these feed and bleed solenoids. There may be some approximations possible, but I cant think of how to go about it at the moment, because of the significant temp effects.