That seems so foreign to me.
100% of my users will be on Windows. Can't I read the ProcessID when the page loads, then I detect an onunload event start a timer that will fire in 30 seconds. When that timer fires it will check if the ProcessID still exists and if not, the page has closed.
I don't know if that could ever work but I am trying to get your mental juices flowing about what I could do if I really really wanted to be notified when the browsers has closed. And to demonstrate I am willing to be a little unconventional about how I accomplish this.