I applied for PhD but I have not been accepted, and now I want to apply for another university and I want to request recommendation-letter from the same professors, what should I write in my email to the professors to request recommendation again!
Asking for more than one recommendation is perfectly normal. Unless you've given them specific details about a particular position, once a reference is written it isn't too hard to send it out again. However, it is polite to keep the effort required from your referees to a minimum. It is quicker for them to send out 5 references at once, than to do one a week for 5 weeks.
Exactly what you ask for from your referees will depend on how references are requested - are they actively sent with the application, or will the place applied to write to referees? Even with the latter, one of my referees said they needed explicit instructions to respond to each.