This has been bugging me for a long time. Several years ago as my girlfriend (at the time) and myself were about to leave her apartment for the evening. I turned off a fan as she was going out the door.
She: Leave that on!
Her: Because it's hot.
Me: But we're leaving.
Her: It's still going to be hot.
Her: So leave it on.
Me: Oh-h-h-h Ka-a-a-ay.
So I turned the fan back on. But I'm not the kind to just give up on a mystery. Why did she want the fan left on? Could it have been a communications failure? She was from the Philippines and we sometimes had cultural or linguistic misunderstandings. (she had the cutest accent though. I still get turned on when I hear Filipinas talking). I couldn't figure out how that could possibly have been a misunderstanding so finally, as I was driving and she was sitting next to me in the car, I began a careful exploration
Me: So. Why did you want the fan left on?
Her: (exasperated) Sweetheart! It's HOT!
I knew from her tone that I was treading on thin ice but I was too curious to drop it. I entered girlfriend-diplomacy mode:
Me: I know sweetheart (use pet name), it's really hot (agree with her). I get really uncomfortable too (empathize) and I'd really like to keep you from getting hot (be on her side), except it's too late, you're already hot (flatter her).
She slaps me on the arm with the last clause and a quick glance over verified that she was smiling. She bought it. We were back on thicker ice. OK, ease into it:
Me: I'm just curious, you know how I am (remind her of our relationship). So, what good is it going to do when we leave the fan running?
Her: (slaps me on the arm again, uhoh) Sweetheart! The fan keeps the room cool!
Me: (light dawning) Uh. You mean, you think... (quick backtrack) You mean the fan is going to keep the room cool even when no one is in there?
Her: Of course!
This possibility had quite honestly never occurred to me. I was shocked. Appalled. Don't they teach basic physics in the Philippines? I actually started to try to explain her misconception. I was old enough to know better but I was just so stunned that I lost my highly refined sense of girlfriend diplomacy for a moment. This episode has gnawed at me ever since, and recently exploded into full-blown frustration when I found someone else leaving a fan on in a closed, unoccupied room.
So allow me to point out that FANS DON'T COOL THE AIR! All fans do is make the air move. Moving air can cool you off if your body is hotter than the air and/or if your skin is moist (and for a healthy person the skin is always a bit moist). The only thing a fan running an empty room does is HEAT THE AIR! Yes, every watt of electricity that goes into that fan to make it run, stays in the room as heat.
Thank you for letting me get that off my chest.
UPDATE: Zantar points out another reason to leave a fan running in an empty room: a room with poor air circulation and a fish tank. Of course I could have been more precise and said that a fan will help keep anything cooler if it's warmer than the air or it's wet (assuming humidity is not too high). Fish tanks are probably warmed noticeably by the pump and the light, and some evaporation can happen at the surface, so fans probably help cool fish tanks. A fan can also help if it is near a window or doorway either blowing hot air out or cold air in. Probably, if the fan is near the ceiling, you want it blowing hot air out and if it's near the floor you want it blowing cool air in.
One more thing: if you keep the room closed, the room will eventually reach thermal equilibrium with a saturated atmosphere and then the fan won't help any more. Of course that could take a long time, but still.