Most of the hair dryers that I have gutted for parts had a 12-24V DC motor in them. They used the elements as a voltage dropping resistor to power the motors. These motors simply CANNOT be directly connects to the mains. Take the time to trace out the circuit to find out how your dryer works.
If you pull off the fan blades and pull the motor out of it's plastic holder, you will most likely find a motor number on it.
If you pull off the fan blades and pull the motor out of it's plastic holder, you will most likely find a motor number on it.