When deriving the Streeter$-$Phelps model, we have:

$\frac{d[O_2]}{dt}=k_{O_2}*([O_2{]}_{\text{s}\text{a}\text{t}}-[O_2])-k_{\text{B}\text{O}\text{D}}*BO{D}_0*exp(-k_{\text{B}\text{O}\text{D}})-v*\frac{d[O_2]}{dx}$

This equation $(1)$ ignores the loss of BOD from sedimentation. We can assume this process following a first order kinetic:

$\frac{\text{d}\text{B}\text{O}\text{D}}{d}$ due $to$ sedimentaiton $=-k_D*BOD$

Rewrite equation $(1)$ by adding this BOD sedimentation term.