Insane Hydraulics

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Let's Talk About Hydropneumatic Accumulators Part 3 - Getting Real

Ok, so now, after we've looked at the isothermal and adiabatic behavior of a diatomic ideal gas in a hydropneumatic accumulator (Part 1 and Part 2), let us talk about sizing.

Sizing is important, but it can be quite confusing, especially for less experienced techs, who, like all normal people, can get intimidated by the formulae engineers put in the sizing section of accumulator catalogs. I do not mind the scientific approach, but I always feel that the dry formulae lack "intuitivity".

Sizing an accumulator is finding the correct precharge pressure and the accumulator's size for a given application. My "goto sizing map" is always the same:

application -> pressures -> temperatures -> precharge -> size

Exactly in this order. And it's actually not hard at all.


Before you decide how big an accumulator you should buy (or sell), you need to consider what it's going to do - in other words - how fast it'll be charged and discharged, and what part of the working cycle (charging or discharging) is more important. If it is emergency storage or a closed-loop charge pressure stabilizer - you're "interested" in the discharge cycle, which will happen fast - so you'd be considering the adiabatic discharge line on our interactive graph (don't worry - you don't need to go back to the previous blogs to check it out - there's a new and better one below). If it is a hydraulic spring, that is supposed to create a permanent load, then you can safely consider the isothermal operation. If it is a fast-moving shock absorber, then maybe you should look at the fast-charging adiabatic curve. The point is - you should visualize the operation of the accumulator in a given application, keeping in mind the differences in how a gas behaves when it is compressed (or decompressed) fast vs slowly.


The next thing you need to determine are the maximum and the minimum pressures your accumulator will be working with. This is an important step, and once again it is determined by the application requirements.


You need to carefully consider the ambient conditions, because, as we saw in Part 1, the precharge pressure will be affected by the ambient temperature, and this is especially important for accumulators with higher precharge.


You know the working pressures and you know the temperatures - now it is time to determine the precharge pressure - and this is where you turn your common sense on and see if the generic recommendation for 90% of the minimum working pressure makes sense for the application. The most important thing here is to consider how big the precharge becomes at max. temperature and account for that, because the increase of the precharge shifts the working displacement range closer to the empty position, and you don't want a bladder slapping the poppet valve or a piston hitting the end of travel (possibly at high speed).


And now, knowing the working conditions, and the precharge, you can determine what percentage of the total compressed gas volume will be "working" - and thus, considering the volume of the oil required from the accumulator, you can make an "educated decision" about the size of the accumulator.

Allow me to illustrate with an example, using, once again, our interactive pressure/volume graph, which is now enhanced with two projections, which allow you to set the maximum and minimum pressures and see where their values fall on the three state lines (green - isothermal, red - adiabatic charge, blue - adiabatic discharge). The adiabatic discharge line assumes that the accumulator was charged at ambient temperature to the max. pressure. The recommended displacement range of 25%-90% is highlighted, but this is only valid for bladder accumulators - piston-type hydropneumatic accumulators can, pretty much, "go nuts" with the compression ratios. As before, the top slider sets the pressure ceiling for the Y-axis.

Isothermal and Adiabatic Operation of a Hydropneumatic Accumulator Filled With Ideal Diatomic Gas

Scale, bar
Prech.@20ºC, bar
Temp., ºC
Max, bar
Min., bar

Say, we are sizing an accumulator that will be working as an aid to a closed loop charging system (because we don't want our closed loop to explode due to momentary charge pressure dips), and we decided that our max. pressure would be 22 bar, our min. pressure would be 15 bar, and we've decided that an additional flow of about 100 l/min during 0.01 s will sufficiently supplement our charge pump, so we are aiming at about 1 liter of usable oil, and we know that the ambient temperature will vary from 10ºC to 50ºC.

Let us get the precharge pressure out of the way first, and since this is a "normal" application, let us go with the standard recommendation of 0.9 times the minimal operation pressure, which is the safest bet for a long service life. Let us put the minimum pressure on the bottom slider to 15 bar, the max pressure slider above it to 22 bar, and let us toggle the radio buttons under the sliders so that both of these pressures project onto the isothermal (green) line. A tip - if you are on a mobile device, pulling the sliders may lack precision - but you can tap/click on the gray squares with names/numbers (to the left and right of the sliders) to increment/decrement the values slowly. Now all we need to do is raise the temperature to 50ºC, and then match the precharge pressure so that the minimum pressure projection falls onto the 90% volume mark. Try shifting the precharge slider about and you will see that the correct precharge is around 12.2 bar.

You can also see a couple of other interesting things. The value displayed under the two projected volumes indicates the working volume (i.e.- the percentage of the total volume within the required pressure window). And by shifting the precharge pressure up and down you can see that lowering the precharge pressure makes the working volume window narrower, even though both of the projections stay within the safe limits. You'll get 28.5% of the volume with a precharge of 12.2 bar, and 23.3% of the volume with a precharge of 10 bar. The same happens when the temperature drops, by the way - and you can visualize this by moving the temperature slider.

But wait a minute - the accumulator will be charged slowly, and with a constant pressure supply, but our closed loop will require additional charge flow momentarily, at a high flow rate, and there's nothing isothermal about that - so let us get the temp back to 50ºC, and change the min pressure pointer to the adiabatic discharge line (by using the min radio buttons under the sliders). Now you can see that the available working volume is reduced from 28% to 19%! Please note that this does not change the total volume of oil inside the accumulator, which stays the same, only the amount of oil the accumulator can supply within the required pressure limits (15 bar and 22 bar).

Now, how can we make things even worse? By lowering the temperature, of course! So let us do that - let us lower the temperature to the minimum 10ºC and see what happens. Wow... We are getting only about 17% of the total volume from our accumulator now! And this, my friends, would be the worst-case scenario you should be aiming for if you want the system to be properly sized. If we still want to get a good 1L of oil from an accumulator under the given conditions, it should have a volume of at least 1/0.17 = 5.9L!

You could ask why I didn't choose the pre-charge pressure by pointing at the adiabatic discharge line instead of the isotherm. It's a fast discharge anyway, isn't it? We could bump the precharge pressure up to 13.6 bar, and then "get away" with a 5.3 l accumulator, couldn't we? At least that's what the graphs tell us. Well... You are kind of right, but the real-life line will fall above the adiabatic discharge line anyway, and over-sizing is way better than under-sizing, so I choose the isotherm for the precharge determination as a safer bet. But would a 5l accumulator precharged to 13.6 bar work for this application? It absolutely would. But not as good as the 6L one at 12.2 though.

And what if we precharged it to 10 bar? 8bar? 14 bar? (These are all bar absolute, by the way - don't forget this, even though this only really matters for low pressures). Then move the precharge slider and see what happens. Basically - the system will always work, but the "working widow" on the volume axis will shift about, and your goal, as a technician, is to choose the best place for it, not any place!

Ok... We're done with sizing... Let us get real now. And by real I mean let us look at how the real gas nitrogen behaves under pressure. And as it turns out - at high pressures it behaves quite differently from the ideal gas we've been playing with so far. First of all - the damned thing has the so-called compressibility factor. If you have a liter of nitrogen at 200 bar, and you were to compress it isothermally to 400 bar, you would find that strangely, it occupies 0.6 liters and not the expected 0.5L! And, to make things, worse, it changes dynamically both with pressure and with temperature! But there's more. Our beloved adiabatic exponent of 1.4 is also not a constant for the nitrogen and it also changes with pressure and temperature!

So, how can you make sure that the real gas behavior of nitrogen is accounted for, then?

Theoretically, you can "go scientific" - and apply the known state equations that use empirical constants - like the Beattie-Bridgeman Equation of State (5 constants), or the Benedict-Webb-Rubin Equation of State (8 constants). These are great, but there's a catch - while you indeed can calculate the pressure for a given quantity of nitrogen inside a given volume at a given temperature, calculating the volume is not possible without approximations/iterations, etc... - so you either use a computer program for this, or you'll spend like a weekend doing calculations for a single accumulator case, and in the end will come up with a required accumulator volume of ten to the fifth cubic meters, because you used the wrong units "somewhere along the way"...

Then you can use the known correction factors - and this is a pretty good way to get answers fast. You calculate an ideal figure and then apply a correction factor from a chart to make it real. Pretty much all hydropneumatic accumulator manufacturers publish tables with correction factors - but it is interesting to note that some do it "the other way around". For example, Rexroth, and EPE give you these charts:

the factors are less than 1, and multiplying the ideal working volume by them will give you the real volume. Hydac, on the other hand, gives you the charts with correction factors above 1:

and you multiply the ideal accumulator volume by this factor to get the real accumulator volume.

I, personally, like Hydac's approach better.

And there's a third option - use a known good app for that. Let me show you real quick. With the same example: 22 bar max, 15 bar min, the required supply volume of 1L, and the temperatures between 10ºC and 50ºC.

Hydac's ASPlight produced the precharge recommendation of 12.1 bar and the recommended volumes of 5.5 and 6.3 liters for the temperature range (pretty much spot on with our assumptions from above):

The Stauff's accumulator sizing web app recommended the precharge of 12.2 bar and the accumulator volume of 6.6 liters. Pretty conservative... But I'll take it!:

The Parker/Olaer's ACCU2203 program (which you need to download, install, and register to use - something that in the world of web apps people don't do anymore...) recommended 12.1 bar precharge and 6.17 liters. Pretty close to our interactive graph solution as well:

The FOX's software surprised me with an unexpected 13.5 bar precharge (I guess someone's using the adiabatic line after all), and... refused to accept the temperature values. A total failure! Here's the screenshot of the app next to the Parker/Olaer's window:

Anyhow, if you go for a known app, you'll get similar results no matter which one you choose. I ran the Numbers for 150-250 bar (same volume and temperature range), and got: 119 bar precharge, 5.66 - 7.21 L from Hydac; 122 bar, and 6.09 liters from Stauff; and 122.3 bar and 6.65L from Parker/Olaer:

Using the interactive graph from above and the method that I described you'd get the 122 bar pre-charge and 20% of usable volume, which would mean that the ideal accumulator volume would equal 5 liters, but then applying the factor of 1.2 brings it up to the unofficial "consensus" of about 6 liters, so I guess my interactive pressure/volume graph can be considered a valid accumulator-sizing aid after all...

So, basically - you should use apps for accumulator sizing (or the interactive graph and the correction factors) but now, having read and soaked in all of the above, you will do this more knowledgeably and appreciatively, fully realizing what is happening to the compressed nitrogen inside the accumulator when it is working in the conditions of your particular application!

On a side note - this was supposed to be a short post, but as is always the case, I got "carried away" and a three-paragraph article turned into a multiple-page rant. Today I only considered the fluid storage application. Different recommendations would be used, for example, for pulsation dampening (e.g. the precharge of 0.6 of the working pressure), or thermal expansion, but still - I wanted to make the "gas spring" behavior as visual as possible, and, hopefully, in that I succeed. If you are reading this - I thank you for your patience!