Ending a session on browser close only (firefox)

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ASP.NET

Javascript

I'm trying to end a user's session when they close the browser, because they never logout properly.

I need to change a value in the db after they close the browser only.

<script type ="text/javascript" >
      var clicked = false;
      var xmlHttp
      var browser = navigator.appName;

      function CheckBrowser() {

          if (clicked == false) {
              xmlHttp = GetXmlHttpObject();
              xmlHttp.open("GET", "../LogOut.aspx", true);
              xmlHttp.onreadystatechange = function () {
                  if (xmlHttp.readyState == 4) {
                       //alert(xmlhttp.responseText)
                  }
              }
              xmlHttp.send(null)
              if (browser == "Netscape")
                  alert("Nog een fijne dag.");
          }

          else {

              //alert("Redirected");
              clicked = false;
          }
      }

      function GetXmlHttpObject() {
          var xmlHttp = null;
          try {
              // Firefox, Opera 8.0+, Safari
              xmlHttp = new XMLHttpRequest();
          }
          catch (e) {
              //Internet Explorer
              try {
                  xmlHttp = new ActiveXObject("Msxml2.XMLHTTP");
              }
              catch (e) {
                  xmlHttp = new ActiveXObject("Microsoft.XMLHTTP");
              }
          }
          return xmlHttp;
      }

 </script>

That works for every browser except firefox.

Please help me with this.

Grant