Practice this exercise first without the flute to be able to focus on the mechanics of breathing.
Your throat and mouth should be open while playing to ensure that they are open while breathing in.
Feel that you are playing a note that is 16 beats long in each staff, where the breath is part of the note. Do not tongue the notes (except in the variation that follows). Don't stop the air at any time. Feel that the breathing in and out are linked seamlessly.
Practice the following at different tempo markings. I suggest that you practice at quarter note = 60, 80, 100 & 120 bpm.
Staves 1 & 2: Make sure to use the entire rest to breathe in for the half-note and quarter-note rests.
Staves 3 & 4: Set your metronome to eighth-notes and sixteenth-notes subdivisions. Be mindful that your throat and mouth are open while playing each note to ensure that they are open when you breathe in. That way, you'll maximize the amount of air that you can breathe in and minimize the breathing noise.