|"I have a fan that I've taken from another desktop. Are these specific to the model"|
It depends which fan it is. If it's a graphics card or CPU fan then, yes, they are specific to that type/model of CPU or graphics card usually.
If, however, you are referring to a case-fan (ie it just attaches to the case interior, doesn't sit on a component) it's less critical. You can fit just about any type of case fan you can get hold of. So that may be the way to go, just fit a couple of case-fans (or leave one side of the PC case off as long as kids and pets can't get near it).
But a failing fan isn't the only cause of overheating:
1. The CPU may be overheating because the thermal paste is no longer doing a good job. See this: http://www.hardwaresecrets.com/arti...
2. A build-up of heavy dust deposits inside the PC & on components can disrupt airflow & cause overheating. Fix that with a purpose-made canister of dry, compressed air. wait for a fine, dry day.
Remove the side of the case which reveals all the components. carry it outside & give it a careful blast all around with the dry air canister. You can do it inside of course, but it will save making a dusty mess all over your PC den if you do it outside.