Session name

By default PHP session cookie name is PHPSESSID, but by changing it to by example "my_sess_id" doesn`t change it. How to do it? Code:

$di->set('session', function() {
    $session = new SessionAdapter();
        session_name('my_sess_id');
    $session->start();
    return $session;
});

Request header:

Cookie:PHPSESSID=tje3ovbgfb2393v5hbnmpg6gt7;