Unfortunately, hot-swapping an eSATA drive is a bit more complicated than hot swapping a USB drive.
First off, your BIOS needs to support AHCI (click here for more info on AHCI), and your SATA controller also needs to support it as well. Secondly your OS, needs to specifically support hot plug, and in the case of Windows 7, it wont boot if you change to AHCI after the OS has been installed.
So, In my case I need to update firmware on lots of SATA SSDs and want to do so without rebooting, and without worrying about changing bios settings. So in order to keep things simple, I followed the procedure below.
First, you need to detect your drive. So watch dmesg to see what drive letter is assigned to your new disk upon initial connection.
[86527.985994] sdd: unknown partition table
[86528.012820] sd 8:0:0:0: [sdd] Assuming drive cache: write through
[86528.012823] sd 8:0:0:0: [sdd] Attached SCSI disk
[86528.456281] device label btrfs devid 1 transid 11 /dev/sdd
Then, when its time to remove the disk device to the following. Subsitute your disk device letters.
# echo 1 > /sys/block/sdd/device/delete
Now you are free to swap your disk. No reboot, no bios changes, required.