If we only care about the epilimnion $($surface$)$ layer of a lake and build a one$-$box model for it$.$ An inorganic solid pollutant with diffusion coefficient $D={10}^{-12}c\frac{m^2}{s},$ settling velocity $w=20c\frac{m}{s},$ is emitted into the lake near the surface at a rate of $40\frac{g}{s}.$ This pollutant cannot be degraded by biotransformation. The epilimnion layer of this lake has a surface area of $100m^2,$ and depth of $20m.$ The metalimnion $($transition$)$ layer of the lake is about $2m$ deep.

If the pollutant concentration is measured to be $2\frac{g}{m^3}$ in the hypolimnion $($bottom$)$ layer. What is the steady state concentration in the epilimnion layer?

$3.$ 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.