Fan goes turbo the moment I turn on computer

Dell Dimension 8200. Been a nice quiet computer for 6 years. A few days ago, when I was out, I'm told fan that cools CPU decided to start blowing furiously. Then I'm told computer shut itself down.

I didn't find any thermal messages in system log. Computer boots and runs, but fan blows furiously the moment I push the power button to turn it on.

Ran Dell Diagnostics. It passed all tests. But there wasn't a test for the fan.

If it was a thermal problem, something getting hot, wouldn't it take at least a few seconds for something to get hot to trigger the fan? But this fan blows furiously the moment I turn on the computer.

Could it be a faulty temperature sensor somewhere? I wonder where the temperature sensor is. Is it built into the CPU itself?
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