# How much an unsealed piston will compress air

I am having trouble wrapping my head around the idea of a piston/cylinder type arrangement where the piston has a small area around the outer circumference that would allow air to travel between the front and back of the piston. To clarify, imagine a completely sealed adiabatic cylindrical box with a rigid shaft in the center connecting the ends of the box. On the shaft is a electromagnetic mass "piston", which has a diameter just smaller than the container and is free to move about the axial direction on the shaft. If a force is applied to the mass and it moves in one direction, how are you able to quantify whether or not the air is compressed (in which case the air would act as a non-linear spring) - or if the area between the mass and container diameters would be large enough for the air to escape past the mass and into the are of low pressure on the other side without causing the air-spring effect. Surely, there will be a large gray area where both cases will occur. Again, I am having trouble quantifying how much air would be compressed if any.

I have tried using a sealed case where the pressure difference would be found by using P1V1 = P2V2, than introducing Bernoulli''s equation and Continuity to find the flow rate in the opening if it became unsealed at the new state. This doesnt help much though because the pressure P2 used in Bernoulli is in the case of a sealed container and not one where the air is actually escaping past the piston to some degree as it is compressed. Any help is much appreciated, I will try to come up with some diagrams shortly if that would help.

Simon Bridge
Homework Helper
Welcome to PF;
If I read you correctly, you are thinking of something like the setup in a Sterling Engine?

It can help you decide on a strategy by looking at simpler cases that contain many of the essential elements of the setup you want to model.

A simpler case is the standard ideal-gas-filled piston all sealed, top of piston open to the air, put a weight on top, and open a small valve in the bottom. The weight sinks as air escapes, but the remaining air in the piston is also compressed.

If you are thinking of delivering the escaped air back to the top of the piston - then add a connecting tube rather than leaving open to the air. That will be easier to analyze than working how the "air" flows around the piston.

Note: P1V1=P2V1 only applied for a gas in thermal equilibrium with itself.
Bernoullis equation has to be modified for compressible flow.

Overall, the task you have set yourself is potentially quite complicated.
If it was set by someone else, then you should use the materials they have provided to simplify the problem.

Thanks for the response. I haven't heard of the sterling engine much before, but a first review does not seem to quite fit my problem. It's more of an electromagnetic shaker unit, in a sealed cylindrical case. The problem given to me is more of a side task, but I don't believe it can be simplified any more. I know the dimensions of the unit, the mass of the vibrating object, the displacement of the mass and force needed to displace it.

I'll see if I can't do more with Bernoulli's for compressible flow. I will try to treat it as an ideal gas piston and with a hole in the bottom (being redirected to the top) - but I am still having trouble analyzing how much of the air is compressed and to what degree in the chamber (dependent on the hole size and my external applied force?).

256bits
Gold Member
Morning ullrtech,

If you can figure out the flow of air passing by the clearance of piston and cylinder using Bernouilli then loads of good fortune to you.

This you can model as the flow between two plates, with the fluid subjected to a pressure differential.
Or, as mentioned, you can model this as the flow through a tube, or as just a hole through a section of the pistion. In either case, you are left with determining how that flow through the tube, or between the plates should function.

Fortunately, this has already been done. Several of the flows go by the name Poisieulle, Couette, Hagen-Poiselle,

First paragraph of Wiki has good explanation, along with the rest of t.
http://en.wikipedia.org/wiki/Hagen–Poiseuille_equation

A PDF always is of interest for these types of flows.
http://www.owlnet.rice.edu/~ceng501/Chap8.pdf

and a couple condensed versions for parallel plates
http://www.ae.metu.edu.tr/~ae244/do...mesFay/2003/Textbook/Nodes/chap06/node11.html

good luck
http://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node108.html

Good luck

• 1 person
Q_Goest
Homework Helper
Gold Member
Hi ullrtech. The problem you're trying to solve is quite complex. The way I've always done these types of problems is to use the same basic method used for FEA type analysis. First, you break up the problem into a number of different, easily identifiable systems using discrete control volumes, and iterate through some small time step (basically using Euler's method). I wish I knew how to use something like a Runga-Kutta method for these problems but I've never been able to do that because you don't have a single differential equation.

I would break it up into 4 parts. Start with one volume at the end of the piston that is being compressed and a second volume at the opposite end which is being expanded. Third, I would model the acceleration of the piston (use a free body diagram to pick out what forces are acting on the piston and solve for position, velocity and acceleration) and fourth, model the flow of gas around the piston using the Darcy-Weisbach equation. The Bernoulli equation doesn't apply here because you're interested in the irreversible pressure losses through the annular gap.

The volume at the compressed end has work being done to it so the temperature will rise over time. There will also be heat transfer to the walls. The thermal mass of the walls is likely to be sufficiently large that their temperature doesn't change significantly so you can ignore that but you might consider doing some heat transfer if you really want to be rigerous. Otherwise, ignore the heat transfer and just do a first law thermo analysis on the gas.

The volume at the expanded end is very similar to the compressed end except as pressure drops, it will get cooler instead of warmer. The first law and heat transfer also applies to the expanded end.

The flow through the annular area can be done using the DW equation as long as dP doesn't exceed ~ 20% of the absolute pressure. If it does, you need to break up the pressure drop along the length of the piston to minimize dP to less than 20% and recalculate fluid properties for each incremental section. Note that for an annular area, an equivalent hydraulic diameter can be determined so it can be equated to a pipe or tube. There is also an entrance and exit loss that can be easily accomodated using the DW equation. The method of doing pressure drop through pipe using DW is nicely outlined in the Crane paper #410 which is basically the bible of fluid flow for things like this. Note that you might also look at the thermodynamics and heat transfer of this flow but the change in temperature along the length of the piston is not likely to be significant and it would add a lot of complexity.

Finally, you need to figure out what the piston is doing. You'll have your electromagnetic force acting on it (I'm assuming that is a constant here?) which is resisted by the dP across the piston faces and the inertial mass of the piston itself. If the piston is sliding inside the cylinder you might also add frictional forces to your model.

So basically, you set up a program that analyzes very small time increments and model each of the 4 basic areas, compressed end, expanded end, gas flow and piston forces/dynamics. I've done these kinds of things using a spreadsheet although I know a lot of engineers would frown upon that, but it works well and can be linked to fluid properties databases such as Refprop which comes in very handy to do these kinds of analysis.

I realize this sounds very complicated and you may not know where to even start. I think the best way is to just start writing things down on paper, drawing a control volume around each end and draw a free body diagram of the piston, then start figuring out what equations apply. Once you have some better idea of what the overall model looks like you can start writing the code.

• 1 person
Much appreciation for the replies guys - Some great fluids/thermo memories are coming back.

I'll play around a bit in Matlab and on paper the next few days and follow up. The piston is a slightly more dynamically complicated as there are also springs used to resit the electromagnetic force (which is more of a sinusoidal electromagnetic driving force), but I should be able to model that with the pressure difference across the piston.

Simon Bridge
Homework Helper
It is a very complicated problem.
If the piston is just vibrating, then you may be able to make an approximation depending on the amplitude of the vibrations.
It all depends on the details of what you need to find out and to what end: i.e. how accurate you need to be.
There is not really enough information to help you beyond saying it's very complicated in general and point at some approaches to working something out.

I've gotten a few basics started in Matlab, if anyone wants to review. The three control volumes are broken down into the compressed, expanded, and annular areas. I am sure there are faults but the results seem plausible (except for when Di is much smaller than Do and the gauge pressure in CV1 drops dramatically inducing a large vacuum).

For now I've kept it as one cycle (from the devices starting point to one full compression). The diagram does not show the forces of the spring or electromagnetic force applied.

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how we can calculate force developed on piston if compressed air is power source for engine.

Simon Bridge
Homework Helper
Pressure times area.

Pressure times area.

sir compressed air will expand in cylinder and after expansion pressure will be very low
and i have calculated variable flow rate entering in to the cylinder so i want now pressure due to this expansion and flow rate

Simon Bridge
Homework Helper
That can be arbitrarily complicated - air is quite complicated to start with and there is not enough detail.
You will want to research thermodynamics of air, and processes involving systems that are not closed (gas may be added or removed).

Baluncore
It will depend on how long the inlet valve remains open.

If the piston is connected to a crank, then the torque on the crank will be proportional to Sin(crank angle).
Assume a high compression ratio, say 100:1 = 1% when at Top Dead Centre, TDC. 100% at BDC.

A really simple analysis ignoring heat transfer would be ...
The inlet will open at TDC = 0°, say 1% volume.
The inlet will close at say 5% of cylinder volume.
The pressure during the expansion will then be inversely proportional to volume. You can calculate that pressure from volume.
When the volume = 100% at BDC = 180°, open the exhaust valve.
When the piston returns to TDC, close the exhaust valve and repeat the cycle.

A compound engine would use the exhaust from the first cylinder in a bigger bore second cylinder, then possibly the second exhaust in a third cylinder giving triple expansion.Each piston would operate at a different crank angle.

Thank you sir I think sir you understand my problem

Sir how I can relate that volume with pressure on piston at various position of piston in terms of crack angle

Baluncore