Today I am continuing with my talk about hydro-pneumatic accumulators, and we'll go back to the question I asked at the beginning of last week's blog: "Do you have an idea of how much the temperature inside the accumulator changes during adiabatic operation?" I will answer this question, once again, with the help of an interactive graph, and then we'll do a quick hands-on test to see how closely our ideal gas graph emulates a real-life system.

Few hydraulic technicians realize how big a (momentary) temperature rise (or drop) of the gas pressure inside a hydro-pneumatic accumulator can be during fast operation, when rapidly compressed gas heats up (or rapidly expanding gas cools down) acting like a spring with a "variable K factor".

Once again - the only thing you need to know to do the math is the ideal gas law (pV = nRT), and the adiabatic expression pV^γ = constant (γ =1.4 for a diatomic gas). If you combine these two, you'll see that one can also come up with an "equals constant" expression that includes temperature:

**TV^(γ-1) = constant**

This is all you need to calculate the gas temperature inside an ideal gas accumulator that operates adiabatically (no thermal exchange with the outside world).

Obviously, this calculation is cumbersome to perform with a calculator, to say the least. There are a lot of web apps that can do the gas law calculations for you (a good one here), but, frankly speaking, I do not fancy calculator apps where you need to input given parameters and press "calculate" - they only show but a tiny slice of what is going on. I prefer graphs because they show the whole picture, and so I built one that shows it all - isothermal, adiabatic compression, adiabatic discharge, * and* the respective gas temperatures. Let me explain how it works:

Above the graph, you will find the radio buttons to toggle between **"bar, ºC"** and **"psi, ºF"** (I couldn't let our friends from imperial unit countries down). Below the sliders, there's a check box that allows you to hide the temperature lines. The green lines correspond to the volume and temperature during isothermal operation (PV = const), the red lines show the adiabatic compression (adiabatic charge), and the blue lines correspond to the adiabatic discharge, assuming the accumulator was "loaded" isothermally at a given ambient temperature to a pressure set by the bottom-most slider.

p Scale, bar |
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Precharge, bar @20Cº |
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Amb.Temp., Cº |
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p Stored, bar |

Let us appreciate the magic of Javascript for a second - each of the lines has 500 data points, and you can see how modern browsers have no problems at all doing all the calculations instantly even on mobile devices!

So, looking at the graph you can see how the hot and the cold adiabatic lines "wrap" the isothermal curve. Let us look at how impressive the numbers are. If we set the pre-charge to 5o bar, and look at the curves at a 200-bar scale, we'll see that if we "hit" this accumulator with 180 bar, the gas pressure will jump up to 150ºC! And if we slowly (isothermally) load this accumulator to 180 bar, and then unload it quickly, the gas temperature will drop to -100ºC!

It is also interesting to note that at any point of the "hot" line (adiabatic charging), we can draw an imaginary horizontal and vertical line, and the points in which these lines intersect the isothermal line will indicate either the final volume of the final pressure the gas would reach if we let it cool down to the ambient temperature with and without the "loading supply" respectively.

I realize now that this last phrase sounds convoluted, so let me explain what I mean with another example (once again with our accumulator pre-charged to 50 bar at 20 ºC). If we "flash-load" the accumulator to 180 bar and then cap it off and let it cool down to 20ºC, the pressure will fall all the way down to about 130 bar (the imaginary vertical line from the 180 bar point on the red line intersecting the green line). If, however, we kept the 180 bar pressure source connected to our accumulator, and then let it cool down, we'd see that it would initially compress the gas to 40%, and then would gradually let 12% more of its volume in after cooling down (the imaginary horizontal line at 180 bar).

No way this is true, man! We load an accumulator to 180 bar, and then it drops to 130?! You should test this, I dare you!

And... I actually did. I took a standard 0.75L membrane accumulator and pre-charged it to 50 bar. By the way - it is interesting to note that while the accumulator was at 20 ºC before the pre-charge, its shell temperature rose to 23 ºC after I filled it with nitrogen:

This is so due to our charging set-up - we use capillary test hoses for the gas connection (very convenient due to the fast and compact M16 minimess couplings), so the main expansion happens at the gas valve of the Nitrogen bottle, which indeed becomes cold, but then when the gas travels through the capillary hose and then into the accumulator, it actually heats up.

Then I installed pressure gauges at the gas side and the fluid side, and quickly loaded the accumulator (16 l/min, 3/8'' hose) to 180 bar through a check valve, and then disconnected the fluid supply immediately when the pressures hit 180 bar:

And... - what do you know - the pressure began dropping off immediately, and after some 20 seconds fell all the way down to 140 bar:

And check out the shell "light up " in infrared:

A 0.75-liter accumulator, charged with nitrogen to 50 bar, holds 1.5 moles of gas - about 42 grams, so, obviously, 42 grams of a substance, even when heated to a high temperature, can't heat up a 1-kg steel shell, so, quite obviously, this operation is not purely adiabatic by any means, but still - it interesting to see that the 140 bar falls neatly between the green and the red lines, maybe a bit closer to the isothermal curve, and intuitively this feels right. I would assume that a real gas in a real accumulator would fall somewhere in between the ideal curves, and this is exactly what happened.

The main point of all this is not the math, and not even the difference between the ideal and real gas behaviors, but rather the importance of understanding how slow accumulator operation differs from fast.

Another thought experiment - imagine a single-acting hydraulic cylinder connected to an accumulator - a hydraulic spring arrangement of sorts - and then imagine that you drop a heavy load on it. You would see the rod go in fast, then slow down to an almost stop, and then slowly creep in for a bit - and now you can tell exactly why this happens!

That's it for today, but there's more on the topic coming next week.