For a car modification project, it would be nice to have a function to express the pressure in a volume, as a function of the size of this volume, the amount of air moved by a compressor per unit of time, and the time for how long this compressor has been spinning (assume constant speed). To simplify I will assume that no gas can escape the compressed volume at any given time.

Not that I think this will actually be that useful, I'm just curious how you would express this as something that can at least roughly approximate it. Is it possible to derive something useful from the ideal gas equation? The gas that is being pressurized is air at sea level, taken directly from the atmosphere.

The reason this is interesting at all is that it is claimed sometimes that reducing this volume will lower the so-called "turbo lag" in a turbocharged engine. E.g. on Wikipedia we have:

It makes intuitive sense that "lower overall pipe and intercooler length (size)" is simply the size of the volume defined by the combined set of pipes plus the intercooler itself that the compressor (turbo) needs to pressurize. If this volume is larger, it should take more time for the same compressor at the same speed to pressurize it. I'd like to know what this relationship looks like just for fun, and maybe I will learn something.

Is it oversimplification to assume that no air escapes? At least some amount of air will escape into the engine itself. Is it an oversimplification to assume that the compressor will spin at a constant speed? If the volume is larger, I assume it is easier for the compressor to spin up initially.

It's not clear what you mean by 'some amount of air will escape into the engine itself.' The whole idea of the turbo is to increase the amount of air flowing into the engine cylinders, over and above the amount which can be drawn in naturally.

No, the turbo will not spin at constant speed, unless the engine is running at constant RPM.

'Turbo lag' arises from the fact that when you step on the gas, it takes a certain time for the turbo to increase its RPM to provide the extra cylinder charging to boost power output. The turbo is driven by the amount of exhaust gas output by the engine, such that if the engine is idling or running at low RPMs, only a small amount of exhaust gas is available to turn the turbo. A few seconds after stepping on the gas, more exhaust gas is generated by the engine, which in turn is available to speed up the turbo, which forces more air into the cylinders, which creates more exhaust gas to turn the turbo, etc., etc.

If you reduce the length of plumbing between the air intake of the turbo, thru the intercooler, and thence to the intake valves of the engine, you will reduce the amount of turbo lag, but you won't eliminate it entirely.

I'm not sure what your goal is, or what answers you are looking for. The effect that turbocharging has on engine performance is a little more complicated than simply playing around with the ideal gas equation. This is why engineers study thermodynamics.

Another method used by race cars to reduce turbo lag is to cut off the fuel input for 1 or more cylinders (in a pattern to evenly distibute the work load on the cylinders) while allowing more air to flow through all the cylinders. This increases the out flow from the engine into the turbo at partial "throttle" inputs.

Yes, that is the whole idea. I know how the turbocharger system works. As a consequence of this, the chamber (i.e. the volume) immediately prior to the intake valves of the engine will be pressurized. I am interested in the time it takes to achieve this pressure if you are allowed to play around with the aforementioned variables.

I know the turbo doesn't spin at a constant speed, the question is if this simplification affects the result too much that it can be neglected or not. As long as the difference in turbine RPM throughout the pressurization process is not too big when using a small and a large volume, I don't see why it matters.

Actually there are two different notions that are noteworthy, there is something called "boost threshold", which is where the turbo starts to build pressure at all. The turbo lag is the time from the boost threshold to where the turbo produces full boost.

Yes, this is what I already stated. However I have read that the lag is a lot more sensitive to the turbo itself, and not the volume that needs to be pressurized. This of course makes perfect intuitive sense too, consider a very large turbo, and a very small engine; regardless of the volume that needs to be pressurized the small engine will need to work very hard to spin the large turbo at all. This is not really relevant to this particular question though, I simply assume that the characteristics of the turbo itself will remain fixed.

My goal is just to learn more about this and just for fun have some formula to play around with that lets me approximate how much there is to gain by reducing this volume.

Physics is not my field of expertise, hence why I'm asking about this in the first place, nor is engineering. I know fluid dynamics is complicated. But I have already suggested the ideal gas equation might not be sufficient. However what would be interesting to learn is why it is insufficient. What limitiation(s) are encountered in this case that makes the ideal gas equation completely useless to approximate this?

I think it would be interesting to see regardless how the ideal gas law would be used, if you attempted to use it even when knowing it would be a flawed approximation.

To clarify this point, in the case of some race cars, at some partial throttle pedal position, perhaps less than 20% of total pedal travel, the actual intake valves are wide open and the ECU is cutting fuel in order to limit power, while allowing maximum air flow out of the engine for that amount of power, keeping the turbo rpm fairly high.

The same method was used a few years ago on non-turbo charged Formula 1 cars because they directed the exhaust into the rear diffuser flow, allowing for higher downforce at a lower rear wing angle of attack. (The current rules don't allow for exhaust assisted diffuser flow).

As I read compressor maps, I would disagree. Look at the below map. Say the pressure ratio is 2.40 and the RPM is where about 20 lbs/min of air flow is required. This should correlate to the turbo spinning at 85,400 RPMs. Now increase engine RPM (with the same pressure ratio) and the air flow has to increase. The turbo will still be spinning the same speed even though air flow has increased to 35 lbs/min.

In your example the speed doesn't change much because you picked some operating conditions that are close to the surge limit, and the efficiency of the turbo increases as you let it pump more lbs/min. If you increased the flow to say 50 lib/min with the same pressure ratio, the speed would have to increase.

Running close to the surge limit is not a good idea. If you "step off edge of the cliff" to the left of that dotted line, you may suddenly have no flow at all.