GDay Penguin. Welcome!
Ive implemented a lot of PID control in microcontrollers used for various things, but this is one Id personally steer away from. Doing this with an Arduino means you need to get +5V to the board. Which means a power supply inside Ms. Silvia with the associated minor hassle of making it waterproof etc.
In contrast - the standard PID box you get (1/32nd DIN size) lives off 110-240VAC ;) It also looks nicer than a home-made thing is likely to look - and is pretty cheap. The other nice thing is that they are set up for thermocouples (with cold junction compensation) or PT-100 probes.. Just a thought.
The downside is that it is simply a lot less fun from an engineering point of view...