Sessions will end when the browser is closed. If session.use_cookies is on, then it will keep a cookie that expires when the user closes his browser. If it's off and the transsid is on, you will see ?SESSIONNAME=biglongsessionid at the end of the URL.
Either way, it won't be there when the user goes back. If it is, you've probably done something to make it do that, and you should know that the session will most likely be gone the next time they visit.
I would keep user name in a cookie and some sort of identifiable code (not a password) which you can read and log them in manually.