If the machine is shutting down, most likely it's not because a case fan is bad, but it is possible. On Dells, if there is a case fan issue, when you boot, the BIOS would generally detect it (I've had this happen to me) and alert you with a few beeps and a message. I believe on the E510, it's got a heat sink on the CPU and then a plastic shroud over the top enclosing it with a chassis fan on the back of the case. So, you should be able to power it up and feel it blowing out the back of your case.
You should also check that the fan on the power supply is running. This is a possibility as well - as the power supply heats up too far, it can simply turn off and obviously abruptly shutdown the machine. Locate the exhaust fan for the power supply on the back of your machine, and power it up - you should be able to feel it and/or hear it. If not, then find a compatible power supply to replace it with.