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For some reason the rad fan on my 1990 kicks on at a very low temperature and pretty much stays on the whole time. I think the PO messed around with stuff due to an overheating problem but can't be sure. They had removed the overflow bottle and line to the rad tank. I have since replaced this but not sure what is causing the fan to come on so early, and stay on regardless of temperature.
Also wondered if the fan is supposed to shut down when you turn the car off. In my fiat it continued to run, when you shut down until it had cooled sufficiently, but the sensor was in the bottom of the rad.
The fans are controlled by the ECM normally, but since it's obvious the previous owner screwed around with stuff, now you get to trace the wiring from the fans to see where they go. They should go to a set of relays, and then those relays should be wired to the ECM. Your experience may vary.
If they are wired as stock, then the previous owner installed a chip into the ECM to have the fans kick on earlier. They also will not run without ignition voltage, so no, they won't run with the car off.