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The reason why a fluid needs to be incompressible lies in the fact that most hydraulic equipments work upon Pascal’s law. This law put in simple words states that whenever you increase the pressure on an incompressible fluid which is enclosed within constrained boundaries, the increase is equally felt in every portion of the fluid.
Now just imagine that there is an enclosed area containing an incompressible fluid, one side of which has a lesser surface area, whilst the other side has got more surface area in contact with the container. Now if we increase the force applied on the narrow cross section, the rise in pressure on the broader section as a result of this force would be equally great. But pressure is measured in terms of force applied or experienced per unit area, hence if pressure for the two sections is same, and the area of one section is more, then obviously the force on that area is also more, as stated by the Pascal’s law.
Hence we can see that simply by using this property we were able to gain sufficient mechanical advantage which means that by applying lesser force on one side, we can do powerful things at the other end which require more force. The factor by which is force is multiplied depends on the difference in size between the two sections.
For those of you who get confused over this explanation (and I have seen many people) and say that the output of a system can never be greater than its input, as that violates the basic energy conservation principle, let me assure you that no such laws are being broken here (although how I wish they could be) because though force applied seems to have multiplied, the work done is still the same at both ends. Rather the work done at the other end is slightly less than the input because of frictional and other losses, so just relax since the foundations of science haven’t been contradicted here!
If the fluid to be used in this arrangement would have been compressible, it would simply decrease in volume locally upon application of such a force and it would not get transmitted to the other end, hence the fuss about using incompressible fluids. Obviously a gas cannot be used in hydraulic systems, so the incompressible fluids used are mostly hydraulic oils.
Although water is also incompressible but it is usually considered a contaminant rather than a hydraulic fluid in systems, but recently students of Purdue university claim to have built a small lawn mower which uses only water in all its hydraulic systems. This would certainly be an advantage since water has several environment benefits over the petroleum derived hydraulic oil counterparts.
So if hydraulics works best with incompressible fluids, does that mean to say that compressible fluids are worthless? No, for they form a separate branch of study which comes under pneumatics but we will not deviate from the main issue here.
Read more about Hydraulics:
Hydraulics - The Power of an Incompressible Fluid