There could be several culprits.
The radiator fan could be failing to turn on (fan control module or the fan itself. Possibly the cabin air temperature sensor). This will cause the radiator to not do an effective job of cooling the refrigerant for the necessary heat exchange. It would be most obvious at stop lights, when the engine is warm but the fan isn’t cooling in lieu of what you get while driving; the air coming out of the vents in the cabin would get warmer the longer the car is running yet stationary.
The orifice tube (which is like an expansion valve on other cars) could have its filter clogged. Just buy a $4 one from RockAuto, not the $165 Audi one. Plan on replacing this no matter what.
You may indeed have a leak, but it’s actually pretty unlikely. When you added refrigerant, the canister had a pressure gauge on it, right? Did it say that you had low pressure for the ambient air temperature you were in when you added the refrigerant? If the gauge said it was fine Or close to fine, then adding refrigerant nearly certainly over-charged the pressure in your system. Doing that will cause it to continue to blow warm air because the compressor is unable to achieve a significant pressure differential (and thus also temperature differential) Between the two sides of the air conditioning system. You mention it was empty - do you mean you had no pressure except what matched the ambient? If yes, then you definitely have a leak. Probably a nicked hose or ancient and crumbly o-ring.
The compressor could indeed have failed, which means that it is unable to compress the refrigerant and achieve that pressure differential.
Your radiator could be clogged with the black sludge that aluminum pipes get over time when they are slowly corroded by the refrigerant. I don’t think this is the culprit, but it’s possible.
Your desiccant / drier / accumulator canister could be “full.” Said another way, the desiccant inside of it could be saturated with water from atmospheric air that got into your system. It happens, but generally indicates a leak. Plan on replacing this no matter what is wrong with the ac.
The blend doors within the ducts of the cabin may be stuck. They regulate the amount of air coming from the heater core and the condenser behind the dash. They’re like little servo-controlled damper doors that restrict the airflow from the hot and cold sources to achieve the desired temperature setting. They’re behind the dash and kind of a pain to get at.
The cabin interior air temperature could possible be broken, thinking it’s 20*F in the cabin. Possible, but I wouldn’t bet on it.
You may have failed to provide the appropriate frosty offering to the great and fearsome AC god Freon the Magnificent, so now you are cursed with D-Fens style commutes until you make your penance with sweat, money, or time.
If you work on the system yourself, have a shop evacuate the refrigerant for you (don’t vent that very nasty stuff to the atmosphere, and definitely wear goggles: it will blind you if you get it in your eyes - and freon hates your eyes). It costs me $225 to do it at my local guy’s place. They will pull it out, pull a vacuum for a couple of hours to test the system, and recharge it to the correct amount for you for that price. Once the juice is out, you can open up the lines at will and clean / test / replace. Keep lines that are open covered at the openings with some tape to keep out dust and air. Replace every o-ring you encounter, do not reuse them. They are odd sizes, so a standard ac o-ring kit won’t work: get the Audi A4-specific kit or individual o-rings from Audi / VW dealers or Amazon / eBay. Pick up a bottle of PAG46 oil to lube them.
If you replace any one of these three parts - compressor, condenser, radiator, or accumulator, weigh the one you’re removing down to the gram (kitchen food scales or maybe postal scales are good for this), because they all have some measure of PAG46 oil in them. Your replacement parts will need to have PAG46 oil added to or removed from, as the case may be, to match what you took out. It’s pretty important.
You can chase down the leak, if there is one (confirm the line pressures are equal to atmospheric pressure, first) by doing a brief shot of refrigerant UV dye. You will quickly see where the leak is to be found with a UV light.
Personally, I think you may have a leak and killed your compressor, based on your description. For this, you will need to find the leak and replace the part that is leaking, you will probably need to replace the compressor, definitely replace the orifice tube and the accumulator.
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