Volume question related to piston movement in a hydraulic system

[TrpnBils]

This came up in class today and based on the equations it makes sense, but logically I can't get my head around this.

Let's say we've got a hydraulic system with two pistons:

Piston 1: 2cm²​

Piston 2: 12cm²​

Piston 1 moves a distance of 21cm, which according to A₁D₁=A₂D₂ means that Piston 2 moves a distance of 3.5cm.

Given that the area of a cylinder is the area of the circle * height, that should mean that each piston displaces 42 cm³ of fluid.

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Now assume that in a second scenario, the same two pistons move a different distance than they did the first time. Piston 1 moves 30cm and Piston 2 moves 5cm. When you work out the volumes displaced by each piston this time, it comes out to be 60cm³.

Where does this extra volume come from? In a hydraulic system you're not adding fluid or taking it away, right? Even though the cylinders that the pistons move in are different sizes, there is still a finite amount of fluid between the two pistons since it isn't compressed.

What am I (and the other math/science people in the building) missing here?

Thanks!

 

[zgozvrm]

Given that the area of a cylinder is the area of the circle * height, that should mean that each piston displaces 42 cm³ of fluid.

You meant volume, didn't you?

Now assume that in a second scenario, the same two pistons move a different distance than they did the first time. Piston 1 moves 30cm and Piston 2 moves 5cm. When you work out the volumes displaced by each piston this time, it comes out to be 60cm³.

Where does this extra volume come from?

It's the amount of fluid that is moved within the system. One cylinder displaces 60 cm³, the other is displaced by 60 cm³. It's the amount of fluid that had to be moved (displaced) by the 1st piston in order to get the 2nd piston to move 5 cm. Therefore, the system has to have at least 60 cm³ to start with.

 

[TrpnBils]

Yeah i meant volume there, sorry...

Ok, so does that mean that the numbers I'm getting for volume there are actually the volumes of the "empty" space behind piston #1? In other words, the difference between its starting position and its ending position?

 

[zgozvrm]

Yeah i meant volume there, sorry...

Ok, so does that mean that the numbers I'm getting for volume there are actually the volumes of the "empty" space behind piston #1? In other words, the difference between its starting position and its ending position?

Not exactly...
We don't necessarily know how much "empty space" there is behind cylinder #1 to start with.
The volume is a measurement of movement, or displacement (ahead of, or behind) the cylinder.

So if you move your 2 cm² piston forward 30 cm within a cylinder, it's kind of like filling the cylinder with a longer piston. The area of the cylinder (ahead of the piston) is reduced by that area (which, as you stated, works out to a volume of 60 cm³). Being a hydraulic system, that cylinder was filled with a non-compressible liquid (one that takes extremely high pressures, above operating pressures, to compress). Since the volume of the cylinder was reduced, it can no long contain the same volume of fluid, so the fluid travels through the hose (or pipe) and begins filling the 2nd cylinder. Since 60 cm³ of fluid is being displaced, the 2nd cylinder is being filled with the same volume of fluid, thus moving the 2nd piston an amount necessary to equal that volume (in this case, 5 cm).

So you see, it's not that there is more fluid in the second scenario, it's just that more of the fluid (that was already there to begin with) was displaced.