Thermodynamic Open System Question Regarding Air Reservoirs

Hey Guys, new member here looking for a little help. I'm a freelance Product Designer that has very little experience with Thermodynamics, and has a client that expects this is a lot easier than it really is!

What I'm working with is effectively the same things you would find in your average workshop compressor.
Please see the image below:
[ATTACH=CONFIG]780[/ATTACH]
So I have a 20L container, it has a pressure switch which allows air to enter when the pressure is below 1.5 Bar, and stops the air entering when it reaches 3 Bar. Air is exiting the container at a constant flow rate; it is being regulated so that it provides a constant 1.4 Bar pressure at 48.3 L/min.

When the pressure in the container reaches 1.5 Bar, I want to raise the pressure of the container back up to 3 Bar in 60 seconds.

What pressure and flow rate is required coming in to the container to achieve this?

I feel as though it should be not be difficult to work out, but I've tried reading up on the first law of thermodynamics, open system and steady flow processes, but it has not helped!

If anyone has a clue, I would be very grateful!

Below is a calculation procedure I believe is accurate for solving your problem, but it cannot be presented as the work of a certified registered engineer for commercial application purposes.

Before using any of the below it is important to establish if the 48.3L/ min volume fow is the actual flow rate at 1.4 Bar or is it at some standardized Pressure and Temperature value. If it is at the 1.4 Bar pressure then the below should work. Otherwise, then this flow value should be converted to its value at 1.4 Bar before performing the below calculations. The reason I raise this issue is that here in the USA when doing any flow sizing of components we convert all pressure flows to a standardized scfm value at 14.7 psia and 60 degrees F to eliminate the pressure/temperature factor in sizing components.

What you need to do is determine the air volumes in your tank at the maximum 3 bar and minimum 1.5 bar pressures. Assuming the air temperature is constant at both pressures, The volume in your metric units at each pressure is Vmin = 20L x (1.5+1) / 1 and Vmax = 20L x (3+1) / 1. Then subtract Vmin from Vmax. This will tell you the air volume you will have to deliver and the flow rate required to refill the tank on each refill cycle in 1 minute (60 sec). Next, add the 48.3 L/min outflow rate to this value and you will have the total required inflow. I recommend you use the average pressure (3 + 1.5) / 2 as the reference delivery pressure at that flow rate. If the 48.3 L/min air volume is at the regulator discharge pressure of 1.4 Bar then a tank supply inflow volume rate at the average tank pressure should actually deliver a bit more air volume than required for filling in 60 sec but it should assure you can fill the tank within a maximum of 60 sec. However, if the outflow rate is at 1.4 Bar and you need a more accurate rate then reduce this component of required the inflow rate to 48.3 * (1.4+1) / (Pav +1) before adding it to the tank inflow volume.

If you have any questions or concerns please let know before using this calculation.

Amazing, thanks for this! Yes, the outlet flow rate stated on the diagram is an actual value, and I'm happy to neglect changes in temperature at this point. Oh, and thanks for providing everything in metric, it makes things slightly less confusing than swapping between standards all the time!

[QUOTE=JAlberts;7733]I recommend you use the average pressure (3 + 1.5) / 2 as the reference delivery pressure at that flow rate. If the 48.3 L/min air volume is at the regulator discharge pressure of 1.4 Bar then a tank supply inflow volume rate at the average tank pressure should actually deliver a bit more air volume than required for filling in 60 sec but it should assure you can fill the tank within a maximum of 60 sec. [/QUOTE]
I'm still looking to source a suitable compressor for the tank supply, but I was looking for something that could deliver the required flow (78.3 L/min) at 3 Bar or a bit higher to ensure it wouldn't struggle as it approached that pressure (and also to factor in the typically inflated specifications of many imported compressors). But do you think aiming half way (2.25 bar) on a half decent compressor will still achieve my required refill time?

[QUOTE=JAlberts;7733]However, if the outflow rate is at 1.4 Bar and you need a more accurate rate then reduce this [U]component of required the inflow rate to 48.3 * (1.4+1) / (Pav +1)[/U] before adding it to the tank inflow volume.[/QUOTE]
I didn't quite understand this bit. Is this for if I want it to fill exactly in 60 secs? Is 'Pav' Pressure x Area x Volume?

Yes the last bit you questioned is if you wanted to get closer to the exact 60 sec fill time but even with this adjustment I wouldn't expect the time to be exactly 60 sec. Since actual compressor delivery rates will change with the variations in intake air temperature and even a small amount with barometeric pressure there is no calculation that will guarantee an exact repeatable delivery rate under all operating conditions so the best bet is to design for a rate with a bit of safety factor that will at least guarantee a filling time that does not exceed the 60 second requirement.

As to the compressor sizing, you are probably going to end up with a compressor that, provide a higher delivery than required and, without the pressure cutoff, a higher pressure than you require. The rate and pressure that I have had you calculate is intended to be a minimum to insure you do not exceed the mximum 60 sec fill time required by the client. Ultimately you need to contact a compressor supplier representative, explain your requirements and let them assist you in the compressor selection, that is what they are there for.