If the two businesses are separate trades it's best to keep separate books as the Revenue will want separate calculations for the profits of each trade. That being said, there is nothing to stop you having a number of different trades either as a sole trader or a limited company.
Banks do not take kindly to people running a business through a personal account. If it becomes apparent from the flow of funds through your account that a trade is being carried on then they may well start charging you. They don't like to be deprived of their bank charges.
Regards
Joy