To "use" the OS for finding the drive, one would launch Disk Utility, and it would show all "connected" drives. Is that what you mean by "reappeared"? If that does show the external drive/partitions, then maybe you need to reset a Finder Preference in order to see it on the desktop. To do that, go to the Finder Menu heading (right next to the black Apple Symbol), click on it, select preferences, and then click General. It will say "Show these items on the desktop:", and below that, you'll have 4 choices to check or uncheck. The first two are "Hard Disks" and "External Disks". Both of those need to be checked to see your internal drive/partitions on the desktop, and the same for any/all external drives/partitions that are turned on and connected.
If the one for External Disks is checked, but you still cannot see the drive/partitions on the desktop, then it should be one of two things:
1. The "port" on either your iMac or the drive is "defective/bad". That is why I suggested try another port on either the iMac, the drive, or both.
2. The drive itself is going "bad". For that situation, you can use Disk Utility (assuming the drive shows up there) to examine/repair the drive.
(I have read that sometimes simply restarting your Mac can resolve the issue, but I have never needed to do that).
Western Digital drives are notorious for not playing well with Macs. I have a 1 TB External drive, with a Seagate mechanism inside, connected to my Mac Mini via Firewire 800, and it shows up on the desktop (it also shows up on my MacBook Air desktop when I do my backups on Saturdays (that connection is via a Thunderbolt-to-Firewire 800 adapter/connector). I have another 1 TB External Drive, again with a Seagate mechanism, inside a recently purchased iomega enclosure, and that has two Firewire 800 and 2 USB 2.0 ports on it. I only use it when I do my backups on Saturdays, once again via a Firewire 800 connection (or the "adapter" connectivity with my MacBook Air).
I also have a slim, external case that has a 1 TB Hitachi drive inside, and it has only one USB 2.0 port on it. It does mount on the desktop for either of my machines via USB connectivity.
Finally, are you using Time Machine for your backups, just out of curiosity? I myself use SuperDuper!, but the use of either should not make a difference regarding the visibility of the device/devices.