Your remark is factually incorrect. The chairman serves a four-year term after being nominated by the president and confirmed by the United States Senate.
nominated by the president = at the pleasure of the president. If the president is not at pleasure with the chairman, although the POTUS cannot technically remove the chair during their term (although I am sure Donald Trump wished he could), he/she will just serve out his/her term and then won't get re-nominated (example Janet Yellen, Bernake and etc.). Same thing. Usually Fed chairmen serve at least two terms with Janet Yellen the only exception where she only served one term and didn't get renominated for a second term. Would she ever come back to being Fed chair again? Only time will tell.
