kill user session when user close browser

maybe this can help you!

0) use a script to detect window closing event (use lib or your own).
1) and add you custom Ajax code to sent query to your page that kill session.

example (javascript code):

<script type="text/javascript">
        window.onunload = function() {
            alert("Here Ajax Code to page killsession.aspx");
        };
    </script>

example (killsession.aspx page):

Protected Sub Page_Load(ByVal sender As Object, ByVal e As System.EventArgs) Handles Me.Load
        Session.Abandon()
    End Sub