Hi,
Any of the typical Arduino UNO or Duemilanove boards would be fine; you don't seem to need a lot of Input or Output pins.
Will/can the Arduino be connected to a PC to collect or archive data? Would be good to be able to make graphs with Excel etc.. maybe. I think an IB essay will be expected to handle data well and show results clearly.
I'm happy to see someone doing an IB Extended essay on a real physical problem! Let us know if we can help with questions. My wife was the IB Essay coordinator in Saudi Arabia a couple of years ago and almost every one was paper-in and paper-out. Us Engineers think that's BORING... Where are you located?
Take a look at relays and power on the http://ArduinoInfo.Info WIKI here: http://arduino-info.wikispaces.com/ArduinoPower
Look at temperature sensors etc here (I don't think you need thermocouple type): http://arduino-info.wikispaces.com/Brick-Temperature-DS18B20
Let us know how it's going...