What is the relation between speed and temperature of a jet engine?

Question

Mathematical relation

Answer 1

Temperatures will depend on the design of the engine and the prevailing air pressure and temperature (these will be lower at altitude). Less fuel will be required and lower temperatures will prevail for a given speed of rotation at high altitude. For a given set of conditions an empirically derived curve may apply which will take account of factors such as different blade and stator performance at different operating conditions. No such thing as a mathematical formula derived from any thermodynamic principles I'm afraid.