I have the same issue as this with the same MacBook Air. It occurs whatever I boot from: a clean install or various other USB boot drives that I use regularly that are no trouble and are vanilla installs with no extras at all. With that in mind I think that the issue cannot be related to software.
In fact as I write I have just run Apple Hardware Test (by starting up with the D key pressed) and the fan is still running flat out, and did so through the test apart from a few brief slowdowns while AHT did some particular test.
It does appear to be a hardware-related issue. This one also has had no liquid spilled—though that was good to consider as I have seen MacBook Pro 13" 2011 with liquid spill and they behave just like this one does.
In Activity Monitor the system CPU Load is 70%, big red bar. The kernel_task process is using 270%. The fan is running at 6500 rpm, and the temps are all way down in the 30's.
I've done further investigation, and discovered some damage to one of the surface mount devices near the battery connector, so this one is a hardware issue. Not sure whether it is related to liquid though. Picture here: Damage to logic board MacBook Air 2013 11" So I guess this is a power-related issue indeed.
I used Macs Fan Control to check the temps and fan speed.