Curiously, it seems as if it’s using your phone for more than just the initial login; note the orange warning in the screenshot below. If your phone dies, so does your WhatsApp web connection.
Note: Instructions for all mobile platforms are provided on the WhatsApp Web page. If you don’t have that option in the app on your mobile device yet, then either you’re not running the latest version or the company hasn’t enabled the new feature for your account. This seems to be a staged rollout of sorts, so if you’re in this situation a bit of patience will help.