Hi,
The UV Dye when it leaks in small amounts, you need to use a UV Light to see it, but when it leaks in large amounts, it will be visible and green
in colour when the oil leaks from a leak spot.
To understand what the thermal expansion valve does, read about how a basic refrigeration system works, because really that is what the cars A/C System is.
To put it in a nutshell. Basically in order to get cooling, refrigerant needs to turn from a liquid into gas, get a deoderant bottle, all liquid inside, spray it on your hand and notice how cool it gets? That is because of the state change from gas to liquid.
In the refrigeration system that is what happens, the compressor compresses the gas, and in the condenser stage (at the front bumper area), it gets cooled by airflow and it condenses (becomes a liquid), This liquid resides in the receiver/drier and liquid line, you can call this the "waiting room". The Thermal Expansion Valve meters the correct amount of liquid into the evaporator.
Once the liquid goes inside the evaporator (inside dash), this is where it boils off, and turns from liquid to gas, making the evaporator colder. Your TYPICAL expansion valve has a bulb on it that attaches to the low side pipe (larger a/c pipe near firewall), and monitors the temperature. The bulb is filled with a gas, and this gas helps modulate the TXV by pulling or pushing on a TXV, so if the low side line is warm, the TXV opens up more, and meters more liquid into the evaporator, if the low side line is cold enough, the TXV meters less liquid in.
When a TXV valve is blocked with debris or shut closed for whatever reason, it will not be able to meter the correct amount of liquid inside, so not enough evaporation and cooling can happen. If the TXV is not modulating (Stuck Wide Open), it will meter too much liquid, and as a result not enough evaporation can occur as the evaporator is flooded full of liquid.
Proper operation of the TXV will decrease over time, so on second hand cars, especially our nearing 20 year old VR/VS's, a new TXV is a good investment. If your hell bent on getting your A/C working as best as possible, new TXV, New Compressor, New Condenser, and New Reciever/Drier (Mandatory).