Total force from air spring model (double acting piston)

[T-osu]

TL;DR Summary

Trying to calculate total force of air spring modeled as double acting piston. Think I have change in pressure, but having trouble determining change in temperature in each chamber. Very difficult as I am an electrical engineer with no experience in thermodynamics! Any help appreciated.

Hi everyone,
I'm an electrical engineer working on making a linear model for a power take-off system. I've gotten inertial, friction, and hydraulic/electric components done, but what is really confusing me is the gas system; I haven't taken ANY thermodynamics. To simplify it, it is modeled as a double acting piston. So, there are 2 chambers each filled with 1 mole of nitrogen and a piston that initially separates them in the middle with dead volume in each chamber. The rod of the piston is free to move by an outside force. I would add a photo, but I don't see an option to?
The following is assumed:
- piston casing is perfectly rigid
- moving interface is perfectly sealed
- chambers are a closed system, no material exchange with surroundings
- gas pressure and temp homogeneous in each chamber (I know what this means, but not how it affects equations )
- no friction between rod and cylinder
- nitrogen obeys ideal gas law

The gas pressure, volume, and temp change by the energy balance equation (unsure how to utilize this though).
But I am totally stuck...
If I assume that the temperature stays the same, and the piston rod moves by some mass, then I worked out the total force as:
sum_forces = F_lower - F_upper - F_gravity
sum_forces = P_low*A_low - P_up*A_up - m_piston*g
sum_forces = (nRT)*A_low/V_low - (nRT)*A_up/V_up - m_piston*g
sum_forces = nRT [ A_low/V_low - A_up/V_up] - m_piston*g
sum_forces = nRT [ A_low/(A_low*h_low) - A_up/(A_up*h_up)] - m_piston*g
sum_forces = nRT [ 1/h_low - 1/h_up ] - m_piston*g

Would that seem right if we assume temperature stays the same? I realize I forgot to account for dead volume, but I assume I'd just add it in.

Next, I'm working on how to account for temperature change, which is giving me a hard time. So far I've found other posts such as https://physics.stackexchange.com/questions/245808/change-of-temperature-of-gas-in-cylinder, but again I'm unsure. I feel like I need to use the energy balance equation somehow but I can't find any examples.
Could anyone provide any guidance?
Thank you!

 

[Baluncore]

Welcome to PF.

Depending on the speed of operation, changing temperature may or may not be important. I have seen no time parameters for operation.

If compression is rapid, the pressure and temperature will be elevated immediately. If compression is slow, heat will be radiated as it is generated. You will need to estimate the rate of heat transfer from the compressed gas to the environment.

 

[berkeman]

Is this for schoolwork, or for your workplace?

 

[Chestermiller]

You need to be made aware that a gas obeys the ideal gas law (or any real gas equation of state) only if it is at thermodynamic equilibrium (not deforming at finite rate). Otherwise, viscous stresses (which depend on the rate of deformation) contribute significantly to the force per unit area on the piston, and must be taken into consideration. This involves treating the gas forces as non-homogeneous, and requires a fluid mechanics analysis based on partial differential equations.

 

[T-osu]

Hi everyone, Oh I was not expecting replies until after holiday weekend, thank you for quickly responding.

@Baluncore Ah, I see. I expect it to be "slow" based off how the system has been described to me. Also the fact that I've read that there have been experimental tests done and the equation Q_i^dot = K_i (T_amb - T_i) describes the heat transfer between chambers (so now I know yes, I should account for heat transfer), where K_i has been determined for each chamber via these experiments. But I see that I would still need to determine T_i.
So excuse my ignorance as I'm only working off of what the equations tell me, but where does the transfer of heat, Q, come into play? The energy balance equation has del U = Q - W, but wouldn't I just need the ideal gas law to determine total forces as I tried above, and if I know T_i this would allow that? Maybe some pointers to examples of the ideal gas law used in conjunction with the energy balance equation would set me on a good path.

@berkeman This is for my workplace

@Chestermiller The assumptions I have are from the folks in charge of the project, so I am to assume it is to obey the ideal gas law.

 

[jrmichler]

I might be able to help because I have some experience with fast moving air pistons. But first, I need more information. A simple sketch of your system. A description of what is moving what, and how fast. Approximate sizes and strokes - a 0.1" diameter piston is not the same as a 10" diameter piston, depending on velocities and displacements. What is the steady state pressure? Do you know the peak pressures, or is that what you are trying to find? Does the air spring bottom out at either end of the stroke? Is one of the project goals to bottom out without slamming or bouncing?

 

[T-osu]

@jrmichler , yes I attached a photo. I will keep things a little vague as this is an ongoing project.
More generally what is happening is the piston rod is attached to a device that moves it in an irregular sinusoidal heave motion (emulating ocean waves) so everything is moving pretty slowly actually. The air spring's purpose is to act as a restoring force. Estimate speeds of ~ 1 m/s.
Although slightly different, the piston diameter is ~0.12 m in each chamber. The total stroke is ~ 2 m. Initial starting position is ~ 1m.
I don't know the steady state pressure or temperature, but I do know their initial values, upper chamber ~ 4 bar and lower ~ 11 bar. Initial temp in each is 283 K

The goal is to get the force from the air spring throughout time to determine the total force acting on the piston rod (there are multiple forces acting on the piston rod, the air spring is one of them).
If it's not possible to make a linear model of this without making more assumptions (like constant temperature), then that is fine too. I just want to make sure I'm checking everything. Thanks for any help

 

[jrmichler]

Since this is a real world problem, we start with the simplest practical model, then add complexity only if necessary. Even then, we only add enough complexity to make the model work "good enough". Since it is slow moving, ignore temperature. Ignore piston and rod seal leakage.

Then $Pressure * Volume = Constant$. You know the volume of each chamber from the piston position, and you know the pressure at the starting volume. Therefore, you can easily calculate the pressure at any other piston position. Note that the piston rod area reduces the area of the piston on the rod side. The rod force, $F_{airspring}$, is the lower chamber pressure times the rod side piston area minus the upper chamber pressure times the upper side piston area. It's pulling the piston rod upward at a little less than 8000 N in the starting position shown.

As a cross check on your simulation, the rod force should approach infinity when the piston approaches either end of its travel. In practice, something will get destroyed before that point, so the system needs to be designed so that destruction does not happen.

Your finished model will likely be more complex that this. Start simple and get it to work before even thinking about a more complex model.

Since this is an engineering problem, I'm moving it to the Mechanical Engineering forum. The difference between an engineering problem and a physics problem is that engineers want answers that are "good enough", while physicists want completely accurate analytical solutions.

 

[T-osu]

@jrmichler Thanks for moving it to the appropriate forum.
Ah, yes, so I think I was originally heading in the right direction, but it is nice to get some feedback :]
I have a simscape model to compare my more simplified forces to, and so far the force from the airpspring between F_simple and F_simscape compare OK? I guess I would need to see how a difference of 1200 Pa (the largest difference between the two signals) has on the entire system and if ~1000 Pa is even a big difference or not in the world of pressures. (We are working towards simplifying things b/c eventually the complex model takes e-20 time steps, too long to wait).
Would things like heat transfer, leakages, friction, be added for more complexity? I would still like to account for heat transfer, but am unsure how to tackle that.