Controlling the temperature of mica band heaters mounted on extruder barrel very precisely with power rating of 5kW
Contactors are usually acceptable for this application. The current is 22A so you can use a 40A or 50A contactor. Check the life curves for when preventative maintenance should take place. 

Temperature sensor should be a type K thermocouple (maybe type J if you're in the U.S.) and autotune PID control with overshoot inhibition would be normal. Mica heaters are somewhat loosely coupled so they tend to overshoot more than better types, and that can degrade the plastic in the barrel.  

Normally there are several zones on an extruder barrel with each controlled separately and you may need air or water cooling to maintain the temperature accurately when the plasticizing energy is added, which can add some complexity as the dynamics of heating vs. cooling tend to be quite different.