Not sure how I could state the formula so I will just give the problem.
Suppose I have a pneumatic cyl and piston with a 24" stroke, say the cyl is 4" ID. I am trying to figure out a way to calculate the size of a relief hole that would allow the cylinder a full stroke in 1 second with say 100# pushing on it. This is for an exercise machine so the size of the whole and pressure required would be adjustable. I plan on just physically calibrating it using weight but would prefer if I could apply some kind of formula.
The force will be the piston area multiplied by the pressure.
The pressure it takes to get air to go out a relief hole will depend on the diameter of the hole and on the air velocity. It is simpler if the absolute pressure on the low pressure side of the hole is less than 52% of the absolute pressure on the high pressure side of the hole. In that case you get sonic velocity through the hole, for whatever the upstream pressure is.
If the pressure downstream of the hole is more than 52% of the upstream, it’s easier to figure that you get a pressure difference of 1 " water column when the air is moving at 1000 feet per minute, and the pressure is proportional to the square of the velocity.
All this assumes the entire area of the hole passes full flow. If the entrance to the hole is sharp, that is if the hole is a cylinder entering a plane, then the effective area of the hole will only be 65% of its geometrical area, a phenomenon known as a “vena contracta”.
Interesting design. That is going to be one expensive and heavy machine unless you are making this out of PVC pipe or something. Also, if this is a “real” industrial cylinder I don’t know if 100 pounds will even move it let alone in one second. I remember the bicycle stand/exerciser I used to have. It used a 3" diameter fan driven by the tire. Spinning that tiny blower was all a human could do.
If you want 100# of force on your 4" cylinder, you’ll end up with about 8 psi of pressure (above ambient). You can’t have this for the entire stroke - the first part of the stroke will be consumed just compressing that air up to 8 psi (22.8 psi absolute). According to this thermodynamic process calculator, you’ll compress the air to about 73% of its original volume before you achieve this pressure - so, you’ll have 73% of your 24" stroke left over which you can deliver 100# of force. This assumes adiabatic compression, and zero flow from your relief hole during this initial phase.
The air at this point will be at 1.36 times atmospheric density, so now we can estimate flow velocity out of the relief hole. The air has been warmed by compression, but not a lot, only about 13%; it will cool by this much when ejected to atmosphere. To simplify the math, I will assume a barotropic fluid (i.e. density is a function of pressure only) so that we can use the corresponding version of the compressible flow equation to calculate velocity through the relief hole:
For your situation:
p_0 = 156.2 kPa
p = 101.325 kPa ρ_0 = 1.632 kg/m^3 ρ = 1.2 kg/m^3
γ = 1.4 (for air)
I got a velocity of 8.83 m/s. 73% of your original cylinder volume is 0.003608 cubic meters, and you want it gone in one second. A volume flow rate of 0.003608 m^3/s at a velocity of 8.83 m/s works out to a flow area of 0.0004086 square meters, or 0.633 square inches. With the vena contracta phenomenon, your actual hole will need to be about 1.03 square inches in area, or about 1.15 inches in diameter. Since these are approximations, you’d want to start with a smaller hole (maybe 0.75"?) and increase as needed after testing.
If you have a larger cylinder bore, you can develop the desired 100# of force with a smaller pressure rise, which means you’ll waste less % of your stroke trying to get to that required pressure. With larger volume (due to increased bore and increased effective stroke), you would then need a larger relief hole to achieve the desired larger volumetric exhaust flow rate at that lower pressure. Note that seal friction on the piston will provide additional resistance, and it will be greater for a larger bore diameter. The magnitude will depend on the actual cylinder/piston you use.
Thanks for the reply. I may go to a 6" cyl. I was planning a piston that did not use a tight seal favoring a more friction free movement. Thinking of felt seals no rubber. Cyl’s will be made of PVC with a long piston to avoid binding.
A fan is another load source I am looking at. Retired and I have a lot of time on my hands. Cyl’s would be cheap as they are low pressure and could be made from PVC, resistance in them would also be low as a positive seal is not required so felt could be used with no rubber.
If OP goes with a 6" cylinder, the pressure involved is only going to be about 3.6 psi. Unlikely to crack a decent PVC pipe in use, and unlikely to be terribly hazardous if it does. The same should not be assumed for high-pressure applications like backyard potato cannons.