How to stop twitter from automatically logging onto main account?

Question From User - Answers in Comments.
I don't know if anyone else has experienced this but I have multiple accounts that I use for twitter and if I'm logged into any of my newer accounts for any length of time, it will eventually assume me logged in as my first account. So I have to log out and log back in again under the newer account. This is getting incredibly annoying. Is there a way to block twitter from assuming the original account?