Several weeks of silence here, for a bunch of reasons that mostly boil down to "being crazy busy." I've got a bunch of physics posts over at Forbes during that interval, though:
-- The Camera Trick That Justifies The Giant Death Star: I busted out camera lenses and the kids' toys to show how you might make the Death Star appear as huge as on the Rogue One poster.
-- How Quantum Physics Could Protect You Against Embarrassing Email Hacks: Using the DNC email leak as an excuse to talk about quantum cryptography.
-- Four Things You Should Expect To Get Out Of College: Some advice for students starting their college careers about what will really matter for long-term success (the time scale of a career, not just a first job).
-- How Quantum Sudoku Demonstrate Entanglement: One of the things contributing to "crazy busy" was the second round of the Schrodinger Sessions workshop, at which I heard a clever analogy for entanglement from Howard Wiseman by way of Alan Migdall, and turned it into a blog post.
-- Is Your Fan Actually Heating The Air?: We talk about temperature as measuring how fast atoms in a gas are moving. Does that mean that a fan setting air into motion is actually increasing the air temperature?
-- Three Tricks Physicists Use To Observe Quantum Behavior: Another post prompted by the Schrodinger Sessions, this one a big-picture look at the general approaches physicists take to doing experimental demonstrations of quantum phenomena.
So, you know, that's a bunch of stuff, all right.
Another way to figure heat gain is figure out the power draw of the fan, and that all that energy goes into heat. Then given the rate of heat rejected and the amount of air passing by in that time you can take the specific heat of the air and figure out its temperature gain.
It would be really cool if I could read these posts at a site that didn't endanger my computer.
An important detail. You wrote
"That means that the collective motion of the air leaving the fan has been scrambled by collisions with air at some distance from the fan, which means it turns into extra random motion, which we might as well interpret as heat."
I know you were writing for the common man, but that random extra motion *is* thermal energy. It isn not heat. Heat is the transfer of thermal energy from one object to another. Here you have converted the collective motion of the air (which is mechanical energy, which is measured by velocity) into random motion (thermal energy, which is measured by temperature).
And it is all just energy conservation, so Lyle @1 is correct about the best way to estimate the effect.