PC fan running full speed - Solved

We have some HP small chassis PC at work that will do that. The problem is a bad electrolytic capacitor on one corner of the mother board. I believe it's 1000uf at 6.3 volts. If you look at the caps you will see several that have the tops pushed up and it's one of those. Have changed 4 so far and has fixed everyone. Did a Google search initially and found it.

Before you go to checking caps, I would check the paste on the CPU heat sync. I have seen them dry out all the time. They still keep the CPU temp down for the most part but they run just hot enough to trigger the fan. Scrape off the old paste, clean CPU and heat sync and add more paste. you do not need that much, just a dab.

Does it boot all the way? I have a PC with a low battery on the motherboard that I have to go into the BIOS and reenter all the settings when it loses power. Sounds like a jet takeoff until all the settings are entered. Someday I should replace the coin cell on the MB...

It does boot fully, remembers all its settings. I had a look at the capacitors, they all "look" OK, no bulging or leakage. I know that isn't the definitive test, though.
I'll see if I can find my tube of thermal compound, and try the heat sink issue.
Any more thoughts??

Had a look at it again today. Previously I had tried restarting the machine numerous times, with no improvement on the screaming fast fan. But I had occasion to unplug the machine from the AC power; when plugged back in, everything is now working normally. As mentioned in my original post, this happened after a power failure, so I'm guessing it may have been a glitch caused by that.
Thanks to all for your suggestions!

Unplugging, and allowing all the loose ones and zeros to drain out is usually one of my first troubleshooting steps with ANY electronic item. Pressing and holding the power button while it's unplugged can speed the process along too.

Your PC may have a 'quick booting' feature. The way that works is it does not really shut down, just goes in to hibernate or similar mode. Unplugging -- a real shutdown.

any time a computer is NOT behaving. full power down AND UNPLUG... 5 minutes.

on laptops, full power down AND remove battery for 10 minutes..