DO

1. CEE 5134 - 1 - Fall, 2007
CEE 5134
Deoxygenation – Reaeration
and the
The Streeter-Phelps Equation
Thomas J. Grizzard
25 October, 2007

2. CEE 5134 - 2 - Fall, 2007
Streeter-Phelps History
• Modeling of oxygen dynamics in flowing waters has
become a sophisticated discipline in the 21st century
• Modern models are built on the work of Streeter and
Phelps on the Ohio River in the first quarter of the 20th
century (1914 – 1925)
• Developed a relationship to predict longitudinal oxygen
profile in flowing waters as a function of:
– Strength of degradable organic matter
• Consisting of mix of background BOD in stream and BOD
introduced by a waste discharge
– Rate of diffusion of oxygen into the water from the
atmosphere

3. CEE 5134 - 3 - Fall, 2007
Who Were Streeter and Phelps?
• Beginning in 1913, USPHS maintained a laboratory in
Cincinnati dedicated to the study of “the manifold
problems of stream sanitation”
– Eventually morphed into the EPA Cincinnati Laboratory
• H.W. Streeter: Sanitary Engineer at the USPHS
Cincinnati Lab
• E.B. Phelps: Professor of Sanitary Science at Institute of
Public Health of the College of Physicians and Surgeons
at Columbia University
• Studies on the oxygen dynamics of the Ohio River were
commissioned in 1914 and 1915 by Surgeon General
W.H. Frost
– In 1925, Phelps dedicates his book, Stream Sanitation, to
the memory of Frost

4. CEE 5134 - 4 - Fall, 2007
Key Historical Publications
• Purdy, W.C., “A Study of the Pollution and Natural Purification of the
Ohio River, Vol. I, The Plankton and Related Organisms,” Public
Health Bulletin No. 133, U.S. Public Health Service, Washington,
D.C., 1923.
• Streeter, H.W. and W.H. Frost, “A Study of the Pollution and Natural
Purification of the Ohio River, Vol. II, Report on Surveys and
Laboratory Studies,” Public Health Bulletin No. 143, U.S. Public
Health Service, Washington, D.C., 1924.
• Streeter, H.W. and E. B. Phelps, “A Study of the Pollution and
Natural Purification of the Ohio River, Vol. III, Factors Concerned in
the Phenomena of Oxidation and Reaeration,” Public Health Bulletin
No. 146, U.S. Public Health Service, Washington, D.C., 1925.

5. CEE 5134 - 5 - Fall, 2007
Effect of Organic Wastes on Stream Ecosystems
• Streeter-Phelps Model: Dissolved Oxygen Sag Curve
– The Streeter-Phelps Equation is integral to most of the
widely used dissolved oxygen models in use today
– Addition of degradable organic matter (BOD) to a flowing
watercourse causes a slow decrease in O2, caused by
heterotrophic metabolism
– Opposing “deoxygenation” is reaeration, which proceeds
at a rate proportional to the concentration deficit (relative to
the saturation concentration)

6. CEE 5134 - 6 - Fall, 2007
Oxygen Sag Effects on Biological Communities

7. CEE 5134 - 7 - Fall, 2007
Components of the Oxygen Sag Curve
Deficit,
Deoxygenation,
and
Reaeration
(x
-1),
mg/L

8. CEE 5134 - 8 - Fall, 2007
Definitions for the DO Sag Curve
Concentration,
mg/L

9. CEE 5134 - 9 - Fall, 2007
Streeter-Phelps Model Assumptions
• Stream behaves as an ideal plug flow reactor (PFR)
• Flow rate, stream cross section and longitudinal velocity
are constant
• Physical, chemical, and biochemical reactions of interest
are BOD exertion and O2 transfer across air-water
interface

10. CEE 5134 - 10 - Fall, 2007
Streeter-Phelps Model Limitations
• Considers only one “sink” for DO – Degrading BOD
– Missing: NOD, SOD, nonpoint sources, algal respiration,
degradation of microbial products
• Considers only one “source” for DO – Atm. Reaeration
– Missing: Algal photosynthesis
• Downstream movement is by advection only (ideal PFR).
– Missing: Dispersion/Diffusion
• Velocity, depth, BOD exertion, and reaeration are
invariant with distance

11. CEE 5134 - 11 - Fall, 2007
1st Order BOD Exertion Relationship
• It has been shown that, under experimental conditions
approximating those prevailing in a stream containing
reserve dissolved oxygen, this reaction an orderly and
consistent one, proceeding at a measurable rate, and
according to the following law:
“The rate of biochemical oxidation of organic matter is
proportional to the remaining concentration of unoxidized
substance, measured in terms of oxidizability.”
- Phelps, Earle B.

12. CEE 5134 - 12 - Fall, 2007
Quantitatively Stating the BOD Exertion Relationship
• As long as oxygen is present, the rate of biochemical
oxidation of organic matter is proportional to the amount
of organic matter remaining….
 
 
 
 
d
d
-1
d
dL
= rate of BOD exertion = -k L
dt
where: L = organic matter remaining at time, t,
oxygen concentration units,mg/L
k = first -order BOD rate coefficient, t
Separate var
d
d
0 0
-k t
0
dL
iables: = -k dt
L
Integrate & apply boundary conditions (L =L @ t = t ):
L =L e

13. CEE 5134 - 13 - Fall, 2007
What does BOD exertion look like?
t
L
L0 Ln (L0)
d
-k
t
d
-k t
0
L =L e

14. CEE 5134 - 14 - Fall, 2007
Summary of BOD Exertion
• As long as oxygen is present, the decline in
BOD remaining (L) is exponential
• If the system is closed, such as in a BOD bottle,
the DO supply is fixed, and no replenishment
from the atmosphere can take place
– This is the case for laboratory BOD measurements
– Allows construction of a quantitative DO budget between
start and the finish of the test
• What’s different about what happens in “the world?”
– System is open
– BOD is exerted, but DO depletion is opposed by
continuous replenishment from the atmosphere

15. CEE 5134 - 15 - Fall, 2007
Oxygen Replenishment by Atmospheric Reaeration
• Reaeration rate is 1st order with oxygen deficit
• Rate coefficient is related to stream characteristics:
– Velocity, Turbulence, Depth
 
 
 
 
r s
r
s
-1
r
s
dC
= rate of reaeration = +k (C -C)
dt
where,
C = saturation oxygen concentration,mg/L
k = first -order reaeration rate coefficient, t
Can also express in terms of the saturation deficit,D = C -C
Givi r
dD
ng: = -k D
dt

16. CEE 5134 - 16 - Fall, 2007
What Does Reaeration Look Like?
DO,
mg/L
r s
dC
= k (C -C)
dt

17. CEE 5134 - 17 - Fall, 2007
Reaeration: O’Connor and Dobbins Formula
• Based on surface renewal theory
– Model is that “parcels” of water are brought to air-water
interface for some finite time period
– Gas exchange takes place only while a water “parcel” is at
the surface
– After moving away from the surface, water “parcel” mixes
with the liquid bulk
0.5 0.5
r r
1.5 1.5
-1 -1
r r
SI: English Customary:
U U
k = 3.93 k =12.9
H H
k (d ), U (mps), H (m) k (d ), U (fps), H (f)

18. CEE 5134 - 18 - Fall, 2007
Reaeration: Churchill et al. (1962) Formula
• Based on empirical studies of reaeration of under-
saturated waters downstream of dams on the Tennessee
River
• Correlated measured reaeration rates with velocity (U)
and depth of flow (H)
r r
1.67 1.67
-1 -1
r r
SI: English Customary:
U U
k = 5.026 k =11.6
H H
k (d ), U (mps), H (m) k (d ), U (fps), H (f)

19. CEE 5134 - 19 - Fall, 2007
Reaeration: Owens and Gibbs (1964) Formula
• Conducted studies of British streams where oxygen was
artificially depleted by sulfite additione
• Combined British and Tennessee River data
– Correlated measured reaeration rates with velocity (U) and
depth of flow (H):
0.67 0.67
r r
1.85 1.85
-1 -1
r r
SI: English Customary:
U U
k = 5.32 k = 21.6
H H
k (d ), U (mps), H (m) k (d ), U (fps), H (f)

20. CEE 5134 - 20 - Fall, 2007
Atmospheric Reaeration
Depth,
(m)
Depth,
(ft)
• Method of Covar (1976)
• Uses formulae of:
– O’Connor & Dobbins
– Churchill
– Owens-Gibbs
• Input stream velocity
and depth of flow
• Select kr (d-1) at
intersection of
flow and depth
coordinates

21. CEE 5134 - 21 - Fall, 2007
Atmospheric Reaeration
• Method of Covar (1976)
• Uses formulae of:
– O’Connor & Dobbins
– Churchill
– Owens-Gibbs
• Input stream velocity
and depth of flow
• Select kr (d-1) at
intersection of
flow and depth
coordinates

22. CEE 5134 - 22 - Fall, 2007
Reaeration Coefficient Estimation from Stream Descriptions
Water Body Description kr
(days-1 @ 20 oC)
Small ponds and backwaters 0.10-0.23
Sluggish streams and large lakes 0.23-0.35
Large streams of low velocity 0.35-0.46
Large streams of normal velocity 0.46-0.69
Swift streams 0.69-1.15
Rapids and waterfalls > 1.15
Source: Peavy, Rowe and Tchobanoglous, 1985

23. CEE 5134 - 23 - Fall, 2007
Temperature Corrections for Rate Coefficients
• Rule of Thumb is that biochemical reaction rates
double with a 10 oC temperature increase
(Van’t Hoff Rule)
• Arrhenius Equation may be used to more
rigorously correct rate coefficients for
differences in temperature:
a2 a1
a2 a1
a2 a1
E(T -T )
RT T
Ta2
Ta2
o
a2 a1
E
RT T (T2-T1)
T2 T1
k
=
k
Over the range of natural temperatures
(0 - 40 C), the product of T T is ~ constant,
and allows us to define:
= e ...and1st equation becomes: k =k
 

e
Arrhenius
Van’t Hoff
Source:
www.wikipedia.com

24. CEE 5134 - 24 - Fall, 2007
Determine BODULT in Stream after Mixing with Discharge
• Construct mass balance on river flow and waste
discharge to get BODULT of mixture:
Waste Qw,
Input Lw
River
Flow
Qr, Lr
Mixed
Flow
Qr+Qw, L0
r r w w
0
r w
3
r
3
w
3
r
3
w
Q L +Q L
L =
Q +Q
where,
Q =Flow inRiver,L / t
Q =Flow of Wastewater Discharge,L / t
L =Ultimate BOD of river water,m /L
L =Ultimate BOD of wastewater discharge,m /L

25. CEE 5134 - 25 - Fall, 2007
Simplified Schematic Representation of Model
• Assume PF and define control volume as a unit rectangle
• Control volume moves downstream at constant velocity
• Determine the initial oxygen content after mixing (L0)
• Compute DO at any time by solving differential equation
for BOD exertion and atmospheric reaeration
Point
Source Discharge
t0
t1
t2
River
Flow

26. CEE 5134 - 26 - Fall, 2007
Differential Equation for Predicting Longitudinal DO Profile
• Identify a single fluid element in the stream
– Constant volume
– Constant velocity
• Write a mass balance for oxygen on the element as
affected by BOD exertion and atmospheric rearation:
– Accumulation = input – output ± reaction
– Since the fluid element is “intact,” there is no flow in or out,
and the mass balance becomes:
 
in in out out
dC
V = Q C -Q C -(BOD exertion) V +(Rearation) V
dt
 
in out
dC
V = 0 -0 -(BOD exertion) V +(Rearation) V
dt

27. CEE 5134 - 27 - Fall, 2007
deox reaer
d r s
s
d
Remember the mass balance:
dC dC dC
V = V + V
dt dt dt
Substitute the rate expressions:
dC
= -k L+k (C -C)
dt
For math convenience, express in terms of deficit,D = C -C:
dD dC
= 0- =k
dt dt
     
     
     
 
 
 
 
r
L -k D
Streeter-Phelps Development, continued

28. CEE 5134 - 28 - Fall, 2007
Streeter-Phelps Development, continued
d
d
-k t
d 0 r
-k t
r d 0
Substitute the BOD ExertionExpression:
dD
=k L e -k D
dt
Rearrange to:
dD
+k D =k L e
dt
The above is of the standard form:
dz
+Pz = Q (which is not separable)
dx

29. CEE 5134 - 29 - Fall, 2007
Streeter-Phelps Development, continued
r r
d
r r r
r d
r
k dt k t
-k t
k t k t k t
r d 0
(k -k )t
k t
d 0
Use the integrating factor :
e = e
Multiplying each term by the integrating factor gives:
dD
e +k D e =k L e e
dt
And....
d
De =k L e
dt

     
     
 
 

30. CEE 5134 - 30 - Fall, 2007
Streeter-Phelps Development, continued
0
0
r d
r
r d
r
(k -k )t
k t
d 0
(k -k )t
k t d 0
r d
0 0
d 0 d 0
0
r d r d
Separate variables andintegrate:
d De = k L e dt
Giving:
k L
De = e + C
k -k
At boundaries of t = 0 andD =D , we have:
k L k L
D e = e + C .... C =D -
k -k k -k
 
 
   
   

   
   
 

31. CEE 5134 - 31 - Fall, 2007
Streeter-Phelps Development, continued
r d
r
d r r
(k -k )t
k t d 0 d 0
0
r d r d
-k t -k t -k t
d 0 d 0
0
r d r d
-
d 0
r d
Applying the integration constant :
k L k L
De = e + D -
k -k k -k
Solving for D:
k L k L
D = e + D e - e
k -k k -k
By groupinglike terms, simplify to:
k L
D = (e
k -k
 
 
 
   
   
   
 
 
 
)
d r r
k t -k t -k t
0
- e + D e

32. CEE 5134 - 32 - Fall, 2007
Streeter-Phelps Development, continued
)
 
 
 
 
 
 
d r r
-k t -k t -k t
d 0
s s 0
r d
Expressingin terms of DO Concentration:
k L
C =(C -D) = C - (e - e +D e
k -k

33. CEE 5134 - 33 - Fall, 2007
Compute the Critical Deficit
• The critical deficit (D) occurs where the rate of change of
D with time = 0 (dD/dt =0)
• May be computed from the original DE by setting the 1st
derivative equal to zero:
d
d
-k t
d 0 r C
-k t
d
C 0
r
dD
= 0 =k L e -k D
dt
Rearrange to:
k
D = L e
k

34. CEE 5134 - 34 - Fall, 2007
Time to Reach the Critical Deficit (Lowest DO)
 
 
 
 
 
 
 
 
 
d r r
d r r
r
-k t -k t -k t
d 0
0
r d
-k t -k t -k t
d 0
d r 0 r
r d
-k t
dD
As noted, the critical deficit is where = 0:
dt
k L
dD d
= 0 = (e - e )+ D e
dt dt k -k
Taking the derivatives and simplifying:
k L
0 = (-k e +k e )- D k e
k -k
Divide by e
 
   
 
   
   
 
0 r d
r
c
r d d d 0
, take natural logs on both sides and solve for t :
D (k -k )
k
1
t = ln 1-
k -k k k L

35. CEE 5134 - 35 - Fall, 2007
Deoxygenation and Recovery in a Flowing Stream
Concentration,
mg/L
CS
Initial Deficit, Da
Saturation DO, CS
DO
Concentration
DO
Deficit
Travel Time or Distance
Critical
Point
tC
0

36. CEE 5134 - 36 - Fall, 2007
A Streeter-Phelps Model Example Problem
• Wastewater mixes with a river resulting in:
– BOD = 10.9 mg/L
– DO = 7.6 mg/L
– Temperature = 20 C
• Deoxygenation constant = 0.2 day-1
• Average flow = 0.3 m/s
• Average depth = 3.0 m
• What is the distance downstream of the maximum
oxygen deficit?
• What is the minimum value of DO?