I am sending another email before receiving any reply and want to apologize for that at the beginning of the second in an official way. How can I say "I am sorry for sending you the consequent emails" in a more professional and proper way?
First, look up "consequent". I do not think it means what you think it means -- seems like you're looking for "subsequent". Although even that doesn't quite fit here.
I would just say "I apologize for the multiple emails, but . . . " and then explain the reason for the additional email (it's important, something else happened, whatever). That's be kind of average office formal in the United States (although we're not the most formal people). Or "I apologize for the additional email, but . . ." works just as well.