A small city $($population $10,000)$ discharges its untreated sewage into a nearby shallow freshwater

lake. A river flows into the lake and another flows out such that the water volume remains constant.

The winds are strong enough to keep the lake water well mixed. Determine the dissolved oxygen

content and BOD concentration of the lake under steady$-$state conditions $($mg $L^{-1}).$ Represent

the lake as a CMFR$.$ Use the first$-$order decay model for biochemical oxygen demand $($BOD$)$ and the

model we derived in class for oxygen transfer from the atmosphere. Ignore any effect of aquatic life

on the dissolved oxygen balance.

DATA:

Lake:

Volume: $V=2{10}^7{m}^3$

Average depth: $d=10m$

Temperature: $T=20C$

River $($incoming$)$:

Flow rate: $Q_r=0\mathrm{.}5m^3{s}^{-1}$

Dissolved oxygen: $D{O}_r=9\mathrm{.}0mg$

BOD content: $BO{D}_r=0mg{L}^{-1}$

Oxygen transfer from the atmosphere to the lake:

Interfacial mass$-$transfer coefficient: $k_m=5{10}^{-5}cm{s}^{-1}$

Saturation concentration, dissolved oxygen: $D{O}_{\text{s}\text{a}\text{t}}=9\mathrm{.}1mg{L}^{-1}$

Waste:

Flow rate: $Q_w=0\mathrm{.}075m^3{s}^{-1}$

BOD content: $BO{D}_w=150mgL{L}^{-1}$

BOD decay rate: $k=0\mathrm{.}2d^{-1}$

Dissolved oxygen: $D{O}_w=8mgL{L}^{-1}$

$($a$)$ What is the steady$-$state BOD content of the lake? $(10$ points$)$

$($b$)$ What is the steady$-$state DO content of the lake? $(10$ points$)$