Oxygen Transport
Basic Equations and Applications
Environmental Hydraulics
Dissolved Oxygen in Water
Necessary for (aerobic) life in water.
Air is dissolved at the water surface and then transported
into the water mass by turbulence and/or currents.
Solubility of oxygen in water described by Henry’s law:
Cs = k p
Cs: the saturation concentration
k: constant at a given temperature
p: the partial pressure of oxygen at the water surface
(the amount of a gas that dissolves in a liquid is
proportional to the partial pressure of the gas over the
liquid)
1
Saturated Oxygen Concentration
Saturation concentration depends on temperature,
pressure, and dissolved salts
Temp Cs
0C
mg O2/liter H2O
0
5
10
15
20
25
30
35
14.16
12.37
10.92
9.76
8.84
8.11
7.53
7.04
Oxygen content in the water < saturation value
⇒ Uptake of oxygen (dissolution) and transport by
currents and turbulence in the water mass
Simple oxygen balance for well-mixed conditions (uniform
conditions over a cross section):
Vol
dC
= k 1 A(Cm − C )
dt
Thin surface layer with
saturated conditions
2
Inflow of oxygen proportional to the deficit in the water
volume under study:
dC
= r (Cm − C )
dt
r:
re-aeration coefficient (depends on flow
conditions and exposed surface area to
water volume
r = 10-5 – 10-4 s-1
Oxygen Consuming Substances
Example: municipal and industrial discharge of organic and
inorganic matter
Characterized through BOD(t) or COD(t)
(biological and chemical oxygen demand, respectively,
expressed in mg O2/liter H2O)
Model of degradation (first-order reaction):
d
BOD (t ) = − K ⋅ BOD (t )
dt
K: degradation coefficient
3
Oxygen balance:
dC
= r (Cm − C ) − K ⋅ BOD (t )
dt
where:
BOD (t ) = BOD0 ⋅ exp ( − Kt )
(solution to a first-order reaction)
Typical Values on Reaeration Coefficient I
Receiving water type
r20 (day-1)
(at 20 deg)
At other temperatures:
r = r201.024T −20
(T: temperature)
4
Typical Values on
Reaeration Coefficient II
Reaeration coefficient as a
function of depth and velocity
Streeter-Phelps Model
Dissolve oxygen sag curve
5
Dissolved Oxygen (mg/L)
Dissolved Oxygen Sag Curve
10
Initial
Deficit (D
(Da)
8
Saturation DO (Dos)
Deficit
6
4
Critical
Point
DO Concentration (DO)
2
tc
2
4
6
Travel Time (d)
8
10
Streeter-Phelps model:
dC
= r (Cm − C ) − K ⋅ BOD0 ⋅ exp ( − Kt )
dt
Solution:
c = cm − k ⋅
BODo
k
⎛
⎞
BODo ⎟ exp ( − rt )
exp ( − kt ) + ⎜ co − cm +
r−k
r −k
⎝
⎠
6
Define oxygen deficit:
D = cm − c
Solution might be expressed as:
D=
k
BODo ( exp ( −kt ) − exp ( − rt ) ) + Do exp ( − rt )
r−k
(Streeter-Phelps equation)
Oxygen Conditions in Lakes
Significant reduction in the oxygen content in the bottom layer
of deep lakes (d>10-15 m) occurs during the summer season.
Stratification prevents oxygen transport to the bottom layer.
7
Oxygen Saturation in Lake Ivösjön
Summer season
Artificial Aeration
8
Example: Lake Waccabuc, New York
Oxygen concentration
less than 1 mg/l
Aeration starts
Case Study: Silverdalens Paper Mill
Objective: estimate the oxygen conditions in the Silver River
downstream Silverdalens paper mill after Mariannelunds
sulphite pulp mill has closed down
Procedure: develop an oxygen balance model and combine it
with measurements in the river to simulate future scenarios
(from study by IVL)
9
Emåns Catchment
Silver River
Silver River
Mariannelunds
sulphite pulp mill
Silverdalens
paper mill
10
Background
Significant pollution from Mariannelunds sulphite pulp mill.
Additional pollution from Silverdalens paper mill (about 700 kg
BOD7/day corresponding to roughly 1/10 of the total load on the
river). Concentration of BOD7 can be 50 mg/l immediately
downstream the mill.
At Lake Hulingen is the concentration of BOD7 around 20 mg/l.
Flow in Silver River varies markedly over the year:
Normal high flow: 18.7 m3/s
Mean flow: 3.8 m3/s
Normal low flow: 0.6 m3/s
Oxygen Balance Model
The Silver river was divided into six stretches from the paper mill
to Hulingen.
Field measurements were done of temperature, oxygen
concentration, and BOD along the river stretches together with
the floating time. A certain ”water mass” was traced along the
river and sampled.
Two different pollutants were studied, assumed to follow firstorder reactions, together with re-aeration. This yields three
equations for every river stretch having different coefficient
values.
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Degradation of pollutants:
For each river stretch
d
BOD1 (t ) = − K1 ⋅ BOD1 (t )
dt
d
BOD2 (t ) = − K 2 ⋅ BOD2 (t )
dt
Oxygen balance model:
dC
= K 3 (Cm − C ) − K1 ⋅ BOD1 (t ) − K 2 ⋅ BOD2 (t )
dt
Solution to pollutant degradation:
BOD1 (t ) = BOD1 (to ) ⋅ exp ( − K1 ( t − to ) ) )
BOD2 (t ) = BOD2 (to ) ⋅ exp ( − K 2 ( t − to ) ) )
Oxygen balance:
Valid for to < t < ti
dC
= K 3 (Cm − C ) −
dt
K1 ⋅ BOD1 (to ) ⋅ exp ( − K1 ( t − to ) ) )
− K 2 ⋅ BOD2 (to ) ⋅ exp ( − K 2 ( t − to ) ) )
12
Solution to oxygen balance:
C (t ) = Cm − ( Cm − C (to ) ) exp ( − K 3 (t − to ) )
−
K1
BOD1 (to ) ( exp ( − K1 (t − to ) ) − exp ( − K 3 (t − to ) ) )
K 3 − K1
−
K2
BOD2 (to ) ( exp ( − K 2 (t − to ) ) − exp ( − K 3 (t − to ) ) )
K3 − K 2
(solution consists of a homogenous part and a particular part)
Field Measurements I
Floating time:
stretch #
time (days)
5
0.036
6
0.37
7
1.17
9
1.39
10
2.33
11
3.41
Flow in river during experiment: 0.5-0.75 m3/s
Flow from paper mill: 0.08 m3/s
13
Field Measurements II
Analysis of samples (average of three):
Stretch
Temp.
Oxygen
BOD7
#
0C
mg/l
mg/l
3
23.6
0.41
9.9
5
21.7
3.87
11.4
6
19.4
2.67
5.7
7
17.0
2.63
7.3
9
16.1
2.55
5.3
10
16.3
2.08
6.7
11
15.7
3.63
4.7
Two different pollutants; one from the sulphite pulp mill and one
from the paper mill.
Measurements during stoppage of sulphite pulp mill (summer
1976). => estimate of K1
Measurements upstream paper mill => estimate of K2
Additional calibration against measurements along the various
stretches (K1, K2, and K3).
=>
Coefficient values from laboratory experiments to low.
Increased values for field application.
Simulations of conditions after closure of
Mariannelunds sulphite pulp mill.
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Laboratory Measurements of BOD1 Degradation
Best-fit line according to first-order
reaction model shown
Laboratory Measurements of BOD2 Degradation
Best-fit line according to first-order
reaction model shown
15
Simulations with Oxygen Balance Model
Example 4: Tutorial on Heat and Oxygen Transport
An oxygen-depleting material (i.e., with biochemical oxygen
demand, BOD) is released to the river in the previous problem
at the upstream measurement point, where the oxygen
concentration is still 4.5 mg/l. The material has a known decay
coefficient of k=0.000011 s-1 and follows a first-order reaction.
Through the effects of the pollution release, the oxygen
concentration decreases in the downstream point to 5.2 mg/l
(compared to previous problem). Determine the initial BOD
concentration for the material. The re-aeration coefficient is the
same as in the previous problem.
Info from previous problem:
15 km between measurment points with U = 0.5 m/s
T = 20 deg
r = 2.1 10-5 s-1
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