Recollements from cotorsion pairs

We consider a complete hereditary cotorsion pair (A, B) in a Grothendieck category G such that A contains a generator of finite projective dimension. The derived category D(B) of the exact category B is defined as the quotient of the category Ch(B), of unbounded complexes with terms in B, modulo the subcategory B~ consisting of the acyclic complexes with terms in B and cycles in B. We prove that there are recollements with central terms the homotopy category or teh derived category of B We will explore the conditions under which exB=B~ and provide some examples. Symmetrically, we prove analogous results for the exact category A.