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.

11

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.

14

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

16