Dissolved Oxygen Depletion

Dissolved Oxygen Sag Curve

Mass Balance Approach

• Originally developed by H.W. Streeter and E.B. Phelps in 1925

• River described as “plug-flow reactor”

• Mass balance is simplified by selection of • Mass balance is simplified by selection of system boundaries

• Oxygen is depleted by BOD exertion

• Oxygen is gained through reaeration

Steps in Developing the DO Sag

Curve

1. Determine the initial conditions

2. Determine the reaeration rate from stream geometry

3. Determine the deoxygenation rate from 3. Determine the deoxygenation rate from BOD test and stream geometry

4. Calculate the DO deficit as a function of time

5. Calculate the time and deficit at the critical point

Selecting System Boundaries

Initial Mixing

Qw = waste flow (m3/s)DOw = DO in waste (mg/L)

L = BOD in waste (mg/L)

Qr = river flow (m3/s)DOr = DO in river (mg/L)Lr = BOD in river (mg/L)

Qmix = combined flow (m3/s)DO = mixed DO (mg/L)La = mixed BOD (mg/L)

Lw = BOD in waste (mg/L)

1. Determine Initial Conditions

a. Initial dissolved oxygen concentration

b. Initial dissolved oxygen deficit

DODOD s −=

rw

rrww

QQ

DOQDOQDO

++=

where D = DO deficit (mg/L)

DOs = saturation DO conc. (mg/L)

DODOD s −=

mix

rrww

sQ

DOQDOQDOD

+−=

1. Determine Initial Conditions

DOsat is a function of temperature. Values

can be found in Table 6.9 (Gilbert Masters)

c. Initial ultimate BOD concentration

LQLQ +

rw

rrww

aQQ

LQLQL

++=

2. Determine Reaeration Rate

a. O’Connor-Dobbins correlation

where kr = reaeration coefficient @ 20ºC (day-1)

u = average stream velocity (m/s)

2/3

2/19.3

h

ukr =

u = average stream velocity (m/s)

h = average stream depth (m)

b. Correct rate coefficient for stream temperature

where Θ = 1.024

20

20,

−Θ= T

rrkk

Determine the Deoxygenation Rate

a. rate of deoxygenation = kdLt

where kd = deoxygenation rate coefficient

(day-1)

Lt = ultimate BOD remaining at

time (of travel downstream) ttime (of travel downstream) t

b. If kd (stream) = k (BOD test)

and

tk

tdeLL

−=0

tk

ddeLk

−=0

tiondeoxygenta of rate

3. Determine the Deoxygenation

Rate

c. Correct for temperature

20

20,

−Θ= T

rr kk

where Θ = 1.135 (4-20ºC) or 1.056 (20-30ºC)

4. DO as function of time

• Mass balance on moving element

• Solution is

DkLkdt

dDrtd −=

• Solution is

( ) ( )tk

a

tktk

dr

ad

trrd eDee

kk

LkD

−−− +−−

=

5. Calculate Critical time and DO

−−−

=ad

dr

a

d

r

dr

cLk

kkD

k

k

kkt 1ln

1

( )Lk ( ) crcrcd tk

a

tktk

ar

ad

c eDeekk

LkD

−−− +−−

=