Automatically eject disk image x minutes after mounting
You can use a cronJob and an apple Script

unMount a disk with AppleScript :

set mountedDiskName to "AirPort Time Capsule"
set diskIsMounted to false

tell application "System Events" to set diskNames to name of every disk
if mountedDiskName is in diskNames then
    set diskIsMounted to true
end if

if diskIsMounted then

    //  put a sleep or a notification here, see explanation below

    log "Disk Found, unmounting now..."
    do shell script "diskutil unmountDisk" &amp; space &amp; quoted form of mountedDiskName

else

    log "Disk Not Found, mounting now…"
    mount volume "afp://AirPort%20Time%20Capsule._afpovertcp._tcp.local"

end if


https://stackoverflow.com/questions/29332873/os-x-applescript-to-check-if-drive-mounted-and-mount-it-if-not

Make a cronJob to run the Script :

* * * * * osascript ~/Desktop/theScriptToBeExecuted.applescript


How can I configure my computer to run an AppleScript at a specific time? Caveat: without using iCal

Put a sleep or a notification in "if diskIsMounted", because if you mount your disk at the cronTab start, your disk will be ejected before the end of your backup ;) 

**I provide you a simple idea and some code to help you, you will certainly have to modify some code.