Flat notes will appear in the same way that sharp notes did previously except that now we will be needing to lower the notes to reduce the distance from one note to the next.
If we take F major as the example, the musical alphabet will work in order until we reach the 4th scale degree. We need a half step between A to B and so we need to lower (or flatten) the B into a Bb to ensure that the major scale formula is correct. Bb is then a whole step below the 5th scale degree and the rest of the scale works.
The key signature of F major is therefore one flat, Bb.