PB - I think there is a fair amount of untrue information, and a whole lot of unknown information being discussed in this thread. I say that based on having been a Task Force Team Leader of HVAC systems at the Hyundai-Kia plant. I didn't enjoy it at the time, but man, it was a tremendous learning experience.
This post might be long, but bear with me. I'm not fully understanding your issue, and will go back as I type this and maybe understand it better. But to me, it seems like you're saying that under certain circumstances your A/C does not engage. First off, part of what I learned in the TFT was that 90% of the time, the reason behind that is from LOSS OF REFRIGERANT. The reason for the loss varies.
You said you put a gauge on it and it was still reading high with the compressor engaged. There is a lot that is not clear with that statement. What pressure do you consider high? Did that gauge show values? I ask because maybe it just showed ranges with no values. Maybe by "high" you mean that it did not drop, like it should have done, perhaps to a green range on that gauge.
And when you said, "engaged," does that mean you pushed the button and it showed the snowflake? Or does that mean you looked at the compressor and could tell the clutch was engaged?
My reply above is not supposed to be accusatory, or arrogant. Sometimes I think it may read that way. But I'm not trying to be that way so please don't try to read it that way.
I will tell you what I think. This is my opinion based on some of the things I've gathered in the thread. I think your system is probably LOW on refrigerant. If your car hit a deer, perhaps your A/C system was effected enough to induce a small leak. Or perhaps it wasn't effected at all. A body shop is not likely going to be the best at properly charging an A/C system.
You mentioned taking it to a shop and having it evacuated. THAT IS A GOOD IDEA. Take it to an A/C repair specialist though. The only, and I mean ONLY way to know that the correct level of refrigerant is in your system is to start with a completely dry system. Evacuate, hold the vacuum to judge if it leaks (if it leaks, the vacuum will work it's way back to no vacuum), charge the system either with automated equipment that weighs it as it charges, or charge it while sitting the refrigerant can(s) on a small accurate scale (how I do it).
2 weeks ago, I had a guy bring his Honda Oddity to my house. We were going to check it out, and if good, evacuate it and recharge it. But, his compressor was dead. Meaning, I would watch the clutch on the compressor and have him turn it on. I would see the compressor clutch engage. And when that happened, my manifold gauges wouldn't even flinch.
And that's one more thing to keep in mind. At steady state, no A/C (compressor) running, the high side and low side pressure are equal. Perhaps this is why some people think their system is overcharged. I'm sure I thought the same before I completely understood how these systems work. I would have expected to see a "low" pressure when connected to the low side. But the pressure in the low side is only "low" when the compressor is working. When the compressor engages (assuming everything in the system is working correctly), from steady-state the high side goes higher, and the low side goes lower. Let's say the A/C system is running and you have the hood up. If you shut off the engine, you will likely hear some hissing for maybe 10 seconds I guess. That hissing is the is the system pressure balancing (low side getting higher, the high side getting lower, until they are equal).
Think of the pressure in the system to act like a seesaw. With the engine idling but the compressor off, the seesaw is completely level. With the engine still idling, but press the A/C button and the compressor starts working, the seesaw tips one way and stays there. One side goes low, and one side goes high, and stays there. Every time that compressor stops, the seesaw levels out.
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