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Oxygen TransportBasic 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 concentrationk: constant at a given temperaturep: 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)

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Saturated Oxygen Concentration

Temp Cs0C mg O2/liter H2O

0 14.165 12.3710 10.9215 9.7620 8.8425 8.1130 7.5335 7.04

Saturation concentration depends on temperature, pressure, and dissolved salts

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):

1 ( )mdCVol k A C Cdt

= −

Thin surface layer with saturated conditions

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Inflow of oxygen proportional to the deficit in the water volume under study:

( )mdC r C Cdt

= −

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 tdt

= − ⋅

K: degradation coefficient

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Oxygen balance:

( ) ( )mdC r C C K BOD tdt

= − − ⋅

where:

( )0( ) expBOD t BOD Kt= ⋅ −

(solution to a first-order reaction)

Typical Values on Reaeration Coefficient I

r20 (day-1)Receiving water type

(at 20 deg)

20201.024Tr r −=

At other temperatures:

(T: temperature)

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Reaeration coefficient as a function of depth and velocity

Typical Values on Reaeration Coefficient II

Streeter-Phelps Model

Dissolve oxygen sag curve

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Initial Initial Deficit (Deficit (DDaa)) Saturation DO (DoSaturation DO (Doss))

DeficitDeficit

DO Concentration (DO)DO Concentration (DO)

CriticalCriticalPointPoint

ttcc

2 4 6 8 102 4 6 8 10Travel Time (d)Travel Time (d)

1010

88

66

44

22

Dis

solv

ed O

xyge

n (m

g/L)

Dis

solv

ed O

xyge

n (m

g/L)

Initial Initial Deficit (Deficit (DDaa)) Saturation DO (DoSaturation DO (Doss))

DeficitDeficit

DO Concentration (DO)DO Concentration (DO)

CriticalCriticalPointPoint

ttcc

2 4 6 8 102 4 6 8 10Travel Time (d)Travel Time (d)

1010

88

66

44

22

Dis

solv

ed O

xyge

n (m

g/L)

Dis

solv

ed O

xyge

n (m

g/L)

Dissolved Oxygen Sag Curve

Streeter-Phelps model:

( )0( ) expmdC r C C K BOD Ktdt

= − − ⋅ ⋅ −

Solution:

( ) ( )exp expom o m o

BOD kc c k kt c c BOD rtr k r k

⎛ ⎞= − ⋅ − + − + −⎜ ⎟− −⎝ ⎠

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Define oxygen deficit:

( ) ( )( ) ( )exp exp expo okD BOD kt rt D rt

r k= − − − + −

−

mD c c= −

Solution might be expressed as:

(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.

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Oxygen Saturation in Lake Ivösjön

Summer season

Artificial Aeration

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Example: Lake Waccabuc, New York

Aeration starts

Oxygen concentration less than 1 mg/l

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)

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Emåns Catchment

Silver River

Mariannelunds sulphite pulp mill

Silverdalens paper mill

Silver River

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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 first-order reactions, together with re-aeration. This yields three equations for every river stretch having different coefficient values.

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Degradation of pollutants:

1 1 1( ) ( )d BOD t K BOD tdt

= − ⋅

2 2 2( ) ( )d BOD t K BOD tdt

= − ⋅

Oxygen balance model:

3 1 1 2 2( ) ( ) ( )mdC K C C K BOD t K BOD tdt

= − − ⋅ − ⋅

For each river stretch

Solution to pollutant degradation:

( )( )1 1 1( ) ( ) exp )o oBOD t BOD t K t t= ⋅ − −

( )( )2 2 2( ) ( ) exp )o oBOD t BOD t K t t= ⋅ − −

Oxygen balance:

( )( )( )( )

3

1 1 1

2 2 2

( )

( ) exp )

( ) exp )

m

o o

o o

dC K C CdtK BOD t K t t

K BOD t K t t

= − −

⋅ ⋅ − −

− ⋅ ⋅ − −

Valid for to < t < ti

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Solution to oxygen balance:

( ) ( )

( ) ( )( )

( ) ( )( )

3

11 1 3

3 1

22 2 3

3 2

( ) ( ) exp ( )

( ) exp ( ) exp ( )

( ) exp ( ) exp ( )

m m o o

o o o

o o o

C t C C C t K t tK BOD t K t t K t t

K KK BOD t K t t K t t

K K

= − − − −

− − − − − −−

− − − − − −−

(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

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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

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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