Originally Posted by Scleros
If the problem is processor overheating, a thorough dusting should have resolved the issue. Also, recent Intel processors will throttle power and/or clock rate before completely shutting down, at which point the heatsink fan should sound like an airplane taking off. AMD processors simply shutdown and/or burnup. You may have another issue, such as a power supply that fails when it warms up/failed power supply fan. How old is the computer? Do you see messages (even briefly) when it shuts down, or any blinking lights?
Bootup into the BIOS setup utility and let the computer sit. That's about as idle as it gets. See if it shuts down in the same time duration. Also verify all your fans are turning.