  Let $(\mathcal{G},\otimes)$ be any closed symmetric monoidal Grothendieck
category. We show that K-flat covers exist universally in the category of chain
complexes and that the Verdier quotient of $K(\mathcal{G})$ by the K-flat
complexes is always a well generated triangulated category. Under the further
assumption that $\mathcal{G}$ has a set of $\otimes$-flat generators we can
show more: (i) The category is in recollement with the $\otimes$-pure derived
category and the usual derived category, and (ii) The usual derived category is
the homotopy category of a cofibrantly generated and monoidal model structure
whose cofibrant objects are precisely the K-flat complexes. We also give a
condition guaranteeing that the right orthogonal to K-flat is precisely the
acyclic complexes of $\otimes$-pure injectives. We show this condition holds
for quasi-coherent sheaves over a quasi-compact and semiseparated scheme.
