Yes, the computer case fan, and the Power Supply fan should run constantly.
If this is in reference to a Dimension 5150, there is one computer case fan in the front. This fan pushes air through a plastic tuned port, (Shroud), onto the Heatsink, which sits on top of the Processor.http://support.dell.com/support/edocs/systems/dim5150/en/sm/index.htm
The above link is to Dell Support, and the Service Manual for the Dimension 5150.
Technical Overview shows the inside of the computer.
Processor's have a thermal limit. Once they exceed this thermal limit, or temperature, the Processor turns off. (BIOS turns it off)
This is a Fail safe feature built-in to keep the Processor from burning up.
Desktop computer fans run all the time, unlike laptops.
However the fans should not run at full speed all the time.
If your desktop computer fan sounds like a 'jet engine' constantly, there may be internal hardware problems, for one.
1) It may be that the thermal paste has dried up.
The top of a Processor's case, and the bottom of a Heatsink are not perfectly smooth.
There are 'Pits, Hills, and Valleys'.
The above mentioned imperfections cause a void. This void is an air pocket.
Air is an Insulator, not a Conductor.
Thermal paste fills these voids, and is an excellent conductor of heat. Transfers the heat from the top of the Processor's case, to the bottom of the Heatsink.
Example of a typical Heatsink/Fan combo,http://www.tigerdirect.com/applications/SearchTools/item-details.asp?EdpNo=3451342&CatId=493
The Heatsink is the finned aluminum looking object under the fan.
(The Dimension 5150 does not use this kind of Heatsink/Fan setup. I included the link to show a typical Heatsink. The front computer case fan, pushes air through a plastic shroud which encases the Heatsink, in a Dimension 5150)
Solution is to Thoroughly clean the top of the Processor's case, and the bottom of the Heatsink, and correctly apply fresh, new thermal paste. (Average cost of thermal paste for this application is $6)
Observe Anti-Static Precautions!]
2) If the inside of the computer is dirty, as well as the inside of the Power Supply, this will also cause the cooling fan/s to race at high speed.
Solution is to use a can, or two of compressed air for computers, and clean the inside of the computer, (Especially the fins on the Heatsink), and the inside of the Power Supply.
[Again, observe Anti-Static Precautions]