Help Calculating time to recharge air tanks
 

This one is making me feel dumb:

I'm trying to optimize the performance of our (rookie team) pneumatics system. One basic parameter is how long it takes the KOP compressor to charge the tanks from the working pressure (60 psig) to the storage pressure (120 psig). I've tried estimating this a couple of different ways, but they all go pretty much like this:

1) Assuming temperature stays constant (not a fair assumption maybe):

P1 * V1 = P2 * V1 (from the ideal gas law)

So ending with a storage volume of 35 in^3 at 70 psig (84psia), to get to start with a larger volume at 60 psig (74 psia):

V1 = P2/P1 * V1
V1 = 84 psia / 74 psi * 35 in^3
V1 = 39.73 in^3

2) and since we started with 35 in^3 of 60 psi air, we need to add:
deltaV = 39.73 - 35 = 4.73 in^3

3) Looking up the performance values for the ViaAir 090c (http://www.viaircorp.com/90C.html#tabs-2), we get a flow rate at 60 psig of
Vdot60 = 0.53 cfm = 15.26 in^3/s

4) So the time to bring the tank from 60 to 70 psig is:
t60to70 = deltaV/Vdot60
t60to70 = 4.73 in^3 / 15.26 in^3/s = 0.3s

5) Repeating this for each 10 psig interval between 60 and 120, I get a total recharge time of about 1.7s per 35 in^3 tank.

Here's my problem - we're seeing much longer recharge times on our robot. With three tanks at above 60 psig, the compressor runs for over 20 s before turning off :confused:

Have I made a mistake in my arithmetic or assumptions? Should I have paid more attention in fluids class? Does ~2s per tank to recharge from 60 to 120 psi sound reasonable in other teams' experience?

(note -I get pretty much the same answer if I just use the viaair flowrate at 60 psig and do the calculation in one interval between 60 and 120 psig)

Re: Help Calculating time to recharge air tanks
 

I'm not entirely sure of how to do this myself, but I don't think your solution accounts for the fact that the amount of gas provided will decrease as the pressure in the chamber increases. The Ideal Gas Law also makes a lot of assumptions that generally don't hold true in the real world so there will be some deviance between the actual time to fill and the expected time to fill. Honestly, I would plot time and pressure in the tank and try to do a regression on it.

Re: Help Calculating time to recharge air tanks
 

I don't think you're accounting for the lower flow rate at higher pressures. If you plot how long it takes to make each 10psi increment you'll find that the higher ones take a lot longer.

Your experimental value of 20 seconds to run from 60 to 120 psi sounds close to what our experience has been in the past. Unfortunately we're not using pneumatics this year so I can't easily get you a comparison value.

Hint for your compressor testing: put a fan on the heat sink. The approved compressors have a very low duty cycle and it's easy to overrun it during tests.

Re: Help Calculating time to recharge air tanks
 

We see much longer recharge times than 0.3 or 2s as well.

I haven't done the calculus, but this is what the curve looks like if you want to play around with it. (It's not an excellent fit, but it's close. Ish.)

Re: Help Calculating time to recharge air tanks
 

I would go through and check all the math, but I learned a long time ago that with air storage it is faster and easier to just plumb it up and try it with a stopwatch to get an accurate measurement of time. I want to say that when we run our systems they charge from empty to full pressure in around 30-40 seconds. That is with reasonable hose runs and 4 stainless clippard tanks. From 60-90 would probably be in the 5-10 second range and 90-120 is at least 10 seconds.

With that said:

1. 2 seconds sounds fast to me, but when it is per tank it isn't as unreasonable.
   
2. as the pressure increases the compressor has to work harder to fill the tank, thus working slower.
   
3. compressed air is a tricky thing to measure with math, and fluid dynamics was never my friend
   
4. you'll likely never see 120 PSI in FRC

Re: Help Calculating time to recharge air tanks
 

Using the function that Excel generated, the integral from 0-120 yields 88.332 seconds for a full charge from 0-120 psi.

Re: Help Calculating time to recharge air tanks
 

This is a bad assumption. The ideal gas law assumes that the energy in the system in constant. The compressor of course puts energy into the system, so that violates PV=PV.

Patrick has the curve for you. It's a calculus problem as he says: Take the integral of the time to fill each step based on the spec.

I'm still a fan of the aforementioned stopwatch method. Or, go to Viair's page, look at the 0.5 gallon fill rates and divide by about 3.3 (0.5 gallon = 115.5 cu in).

Re: Help Calculating time to recharge air tanks
 

I'm tired and don't feel well, which is why I'm not doing the math myself, but something seems fishy to me about just integrating that curve and getting an answer because it's independent of the size of your storage tanks. The integration also doesn't yield time.

You've got volume/minute on the Y axis, and pressure on the X axis, so the integration will yield units of volume*pressure/time, or power (in the charming non-SI unit of inch-lbs/minute)--not time.

Re: Help Calculating time to recharge air tanks
 

Okay, I think I've got it. Integrate the curve between the pressures whose charge time you want, set that equal to the difference in energy of the gas (delta-P*V) over time, and solve for time.

Yes?

(Man, I need a nap.)

Re: Help Calculating time to recharge air tanks
 

This is what I would do. Even this is slightly optimistic as the air will cool off as it flows into, and sits in, the storage tanks. Accurately modeling the compressors performance will be a bit more than trivial.

A decent sanity check would be to take the compressors electrical power input, multiply by a conservative assumption of efficiency, and divide the difference in stored energy by this power to get time.

OP - your initial approach makes too many optimistic assumptions such as no temperature change, constant flow rate vs pressure, etc.

Re: Help Calculating time to recharge air tanks
 

this.

http://www.viaircorp.com/90C.html#tabs-2

lower right corner, "Performance Data" shows 0.34 cfm at 120 psi

It's been my experience with FRC robotics, that if you need to worry about the refill rate of the storage tanks with this much detail, then you're probably using pneumatics for something that would be better handled with a more powerful motor, like a CIM.

Re: Help Calculating time to recharge air tanks
 

Thanks for the responses so far.

...I'm not sure you can just take the integral of that curve. The integral of flow(P) dP would give you something with units of volume/time*pressure (not just time). We would need a function for pressure(volume) for the units to come out right, which is what I was grabbing at (probably wrongly) with the ideal gas law

I only showed the step from 60 to 70 psig. You would need to repeat the next step with the viaair flowrate at 70 psig and so forth for every increment up to 120 psig (where the flowrate is .34 cfm). In fact, the time to fill each increment of pressure goes *down* in my calculation because the pressure increases faster (for a given volume) than the flowrate decreases.

Yeah, when it comes down to it, that's what we'll have to do. I would prefer to have a theoretical grasp of it though, so we could make intelligent choices during the design stage about how many tanks to buy, or how often we can actuate our cylinders.

But it should work! (sigh).

Re: Help Calculating time to recharge air tanks
 

Hooray!

Pro-tip: if your students don't yet know calculus, approximate the curve with a trapezoid (or several) to teach them the *idea* of calculus, and of using unit analysis on axes to determine the physical quantities yielded by integration.

Re: Help Calculating time to recharge air tanks
 

I think the error you are making is that the CFM compressor performance values you are using are at the inlet to the compressor, not the discharge. The compressor does not put out 0.53 cfm @ 60 psi, it compresses 0.53 cfm of atmospheric air (14 psia) when the discharge is @60 psi.

Mike

Re: Help Calculating time to recharge air tanks
 

Yup.

This method is actually how a great deal of engineering calcs are done. Before I knew what I was doing, I brute forced a lot of calculations this way after being inspired by JVN using the method in his drive design calc. Later one I learned better ways to do some calcs, but a lot are still easier to do numerically.

With excel's easy method of repeating formulas, it can be done with plenty precision using small timesteps.

Euler's Method is the simplest (could be wrong...), but there are other methods that can get a lot better, especially if the curve changes direction a lot.