# Insane Hydraulics

## Let's Talk About Hydropneumatic Accumulators (Part 1 - Ideal Gas, Isothermal Operation, Adiabatic Compression)

Today I want to talk about nitrogen-filled hydraulic accumulators, which are used in industrial hydraulics for all kinds of cool things.

I believe that there's no need to explain what an accumulator is, or the details of how a bladder accumulator is different from a diaphragm or a piston one - given the context of this blog we all already know what accumulators are and their typical applications, don't we? What I want to talk about is the science behind the operation of the hydro-pneumatic accumulator, because understanding its basics makes sizing an accumulator (i.e. determining the required volume and/or the correct pre-charge pressure) a knowledgeable and appreciative task.

Let me explain what I mean by that. Accumulators are essentially gas sprigs - which makes them very simple and very complicated at the same time. Simple - when you look at them from the ideal gas perspective and complicated when you take into account all the corrections that you need to introduce in your calculations to account for the real gas behavior of compressed nitrogen. But in this day and age sizing of an accumulator is simple, even in the field - you whip out your smartphone, find an accumulator sizing web app, punch in the numbers - and you're done! There's nothing wrong with this approach - it's practical, and practical is never a bad thing. But I still believe that digging just a tiny bit into the details of a compressed gas behavior arms a hydraulic tech with knowledge that is a) eye-opening and b) useful in many situations:

For example, can you tell me right here and right now how much will pressure drop in an accumulator if the ambient temperature were to drop say by 20Cº? If you can - props to you! But if you can't - you will be in about a minute, I promise! Or what if you have, say, a 20L accumulator with 10 liters of oil "sitting inside it" at 200 bar at room temperature, and your systems "dumps" the 10 liters in 5 seconds - can you ballpark the temperature inside the bladder after this "emergency discharge"? See? That's what I meant when I said "knowledgeable and appreciative".

Now a word about temperature units to my friends from the Imperial Units countries: I am sorry, but we have it easy in Europe, because we use degrees Celsius, which correlate beautifully (1:1) with Kelvins (the only scale that matters!) - all we need to know is that 0Kº is -273.15Cº. But if you are used to the Fahrenheit temperature scale - things can "feel strange". Still - for relative "evaluations" you should bear in mind that one Degree Kelvin increment equals 1.8 degrees Fahrenheit increment, and if you want to go "absolute" all you need is to convert Cº (which is Kº - 273) to Fº by multiplying Cº by 1.8 and then adding the "historical" 32 (no matter how many times I do this, it is still unnatural to me), or, you know, do the right thing - forget about Fº and use Kº, because K stands for "King"!

So, let's jump right into it. Let us imagine that our accumulator is filled with an ideal gas - which is a theoretical gas, that obeys the ideal gas law:

pV = nRT

At relatively low pressures nitrogen behaves very much like an ideal gas - we'll see how it becomes "real" later, but for now, let us continue with the beautiful simple equation from above. The p, V, and T stand for absolute pressure, volume, and absolute temperature, the n stands for the amount of gas in moles and the R is the ideal gas constant that "mates" the units together.

This formula is already very intuitive in the sense that it says - the more stuff you put into a fixed volume, the higher the pressure will be. Since an accumulator is a closed system, in which the amount of gas does not change, we can pretty much assume that the state of the gas inside of our accumulator can be described as "pressure times volume equals temperature times a constant".

Now, let us go back to the question of how ambient temperature changes affect the pre-charge pressure of an accumulator. Since the volume is constant (in other words - our accumulator has the bladder completely expanded) - the formula from above tells us that the p/T or T/p are constant values, which is perfect! Why perfect? Because the p/T is, essentially, bar per degree (in other words - by how many bar the pre-charge pressure will change per degree of the temperature change), and T/p is degree per bar (or how many degrees are required to change the pre-charge by one bar). I went for bar here for convenience, any other units of pressure could be used.

Imagine an accumulator that was pre-charged to 100 bar on a warm Summer day when the ambient temperature was 27Cº (which is the very convenient 300K - don't forget that you need the absolute temperature for these calculations). Now we can state that the pre-charge will rise by 100/300 = 1/3 bar with every additional degree K (or C), or - that it will require a three-degree temperature change to alter the pre-charge pressure by a single bar. So, in Autumn, when the temperature in the shed where the accumulator is mounted drops by 20 Cº (to 280K or 7C), the pre-charge will drop by almost 7 bar! Note that you don't need to know the volume of the accumulator to estimate the pressure/degree ratio - just the absolute temperature and the absolute gas pressure (at any point of its life)! This also means that ambient temperature changes affect more accumulators with higher pre-charge pressures - and now you can instantly tell by how much and impress everybody around you!

Now let us look at how our ideal accumulator behaves when we slowly fill it with oil, so slowly that the temperature of the gas stays constant because there's enough time for the heat generated by the gas compression to dissipate through the bladder and the walls. This is called isothermal operation. The formula from above tells us that in that case, pV is constant, which makes our calculations straightforward - half the volume is twice the pressure. Easy!

pV = constant

Now, I wouldn't be me if I didn't represent this with an interactive graph - so here you have it - the pressure/volume chart of an accumulator filled with ideal gas and working in isothermal conditions. You can use the sliders below the graph to change the precharge (it is assumed that the accumulator was filled at 20Cº), and then change the ambient pressure to see how it moves the isotherm.

### Isothermal Operation of an Ideal Hydropneumatic Accumulator

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

Note that I highlighted the customary 90% - 25% of the accumulator's volume - which is a typical recommendation for a bladder accumulator.

OK, so isothermal operation is simple, but we know that the rubber bladder is quite insulating, so it doesn't "feel right" to assume that the temperature of the gas stays constant - and so we come to another extreme - the so-called adiabatic operation, when we assume that the gas is perfectly insulated from the environment, and its temperature rises when it is compressed (or drops when it expands).

In this case, the math is a bit different, and I won't be detailing that, but the good news is that there's still a constant, only now it looks like this:

pV^γ = constant

p times V to the power of γ equals constant, where γ is the so-called adiabatic exponent, which is 1.4 for diatomic gases (we are aiming for N2, so that's why diatomic). Fun fact - the value 1.4 comes from the thermally accessible degrees of freedom of a molecule, of which the diatomic gas has 5 (3 translational and 2 rotational), but we, obviously, will not be deriving it here. All you need to know is that the exponent will make the pV curve steeper in relation to the isothermal. So, this is how the adiabatic compression plots on our smart pV chart:

### Adiabatic Operation (Compression) of an Ideal Hydropneumatic Accumulator

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

Note that I am talking only about the adiabatic compression. This means that our accumulator is empty of oil and is at the ambient temperature before we begin charging it with oil. There is also "the other end" of the adiabatic operation - the adiabatic discharge - when the accumulator is at ambient temperature already charged with oil, "waiting" for action, so to speak.

And I will discuss this, along with other important points, next week (in part two).