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.

 

[rwisz]

I can understand your confusion about the volume discrepancy in the two scenarios. However, it is important to remember that in a hydraulic system, the volume of fluid remains constant. This means that the volume of fluid displaced by one piston must be equal to the volume displaced by the other piston.

In the first scenario, the smaller piston (Piston 1) moves a distance of 21cm, displacing 42cm3 of fluid. This means that Piston 2, with a larger area, only needs to move 3.5cm to displace the same volume of fluid.

In the second scenario, both pistons move a greater distance, but the ratio of their movements remains the same. This means that Piston 1 moves 30cm, displacing 60cm3 of fluid, and Piston 2 moves 5cm, displacing 10cm3 of fluid. This still adds up to a total of 70cm3 of fluid displaced, which is the same as the first scenario.

The key concept here is that the volume of fluid displaced is not solely dependent on the distance the piston moves, but also on the ratio of the piston areas. This is why the larger piston only needs to move a smaller distance to displace the same volume of fluid.

I hope this explanation helps to clarify the volume question in relation to piston movement in a hydraulic system. If you have any further questions, please don't hesitate to ask. Science can sometimes be counterintuitive, but with a deeper understanding of the underlying principles, we can make sense of seemingly confusing phenomena.