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CE5504 Surface Water Quality Modeling. Lab 5. One-Dimensional Models Thermal Stratification, Vertical Mass Transport and Interactions with Trophic State. Dr. Martin T. Auer MTU Department of Civil & Environmental Engineering. Temperature and the Density of Water. 1.0000. 0.9995. 0.9990.

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CE5504

Surface Water Quality Modeling

Lab 5. One-Dimensional Models

Thermal Stratification, Vertical Mass Transport

and Interactions with Trophic State

Dr. Martin T. Auer

MTU Department of Civil & Environmental Engineering


Temperature and the Density of Water

1.0000

0.9995

0.9990

Maximum

Density

3.94 °C

0.9985

0.9980

Density (g∙cm3)

0.9975

0.9970

0.9965

0.9960

0.9955

0.9950

0 5 10 15 20 25 30

Temperature (°C)


Thermal Stratification

spring

summer

Temperature (°C)




0

1

2

3

4

5

6

7

8

Dollar Bay - Temperature

M A M J J A S O N D

0

1

2

3

Depth (m)

Depth (m)

4

5

6

7

8

Temperature (°C)

0 5 10 15 20



Trophic State and Oxygen Profiles

Eutrophic Lake

Oligotrophic Lake

Temperature (°C)

Temperature (°C)


0

1

2

3

4

5

6

7

8

Dollar Bay - Temperature

M A M J J A S O N D

0

1

2

3

Depth (m)

4

5

6

7

8

Temperature (°C)

0 5 10 15 20


0

1

2

3

4

5

6

7

8

Dollar Bay – Dissolved Oxygen

M A M J J A S O N D

Depth (m)

Dissolved Oxygen (mg∙L-1)

0 2 4 6 8 10 12






Lake Superior – 4/28/00

8

Completely-Mixed Lake


8

8

Lake Superior – 8/25/00

Vertical Resolution - Stratification






X-direction

Mathematics of Diffusion II

Fick’s 1st Law

E

C




Mathematics of Diffusion V

E, turbulent diffusion coefficient, m2/d

E’, bulk diffusion coefficient, m3/d



thermocline heat transfer coefficient

turbulent diffusion coefficient

Application to Vertical Mass Transport


Calculation of the Thermocline Heat Transfer Coefficient

Epilimnion Heat Balance

tributary

heat in

discharge

heat out

surface

heat flux

thermocline

heat exchange

change

in heat


change

in heat

thermocline

heat exchange

upper mixed layer

(~epilimnion)

thermocline

vt∙As

lower mixed layer

(~hypolimnion)

Calculation of the Thermocline Heat Transfer Coefficient

Hypolimnion Heat Balance

assumes no direct heating of

hypolimnion or sediments


uml

lml

Calculation of the Thermocline Heat Transfer Coefficient

Canceling the  and Cp terms, dividing by Vh and expanding …

And defining …

Yields …


Calculation of the Thermocline Heat Transfer Coefficient

Which, for Th at the beginning of the calculation = Thi, yields …

and re-arranging, yields …

further re-arranging, yields …

where a plot of ts versus term

yields 1/vt as the slope

term



0

1

2

3

4

5

6

7

8

Dollar Bay - Temperature

M A M J J A S O N D

0

1

2

3

Depth (m)

4

5

6

7

8

Temperature (°C)

0 5 10 15 20


0

1

2

3

4

5

6

7

8

Dollar Bay – Dissolved Oxygen

M A M J J A S O N D

Depth (m)

Dissolved Oxygen (mg∙L-1)

0 2 4 6 8 10 12


Dollar Bay – Temperature Calculations

For diffusion calculation, need to choose an interval where Tepi is stable and Thyp is warming.

Hypolimnion temperature mass balance.


uml

lml

Sediment Oxygen Demand

Sediment oxygen demand reflects the diffusion of oxygen into the sediment to meet the metabolic needs of microorganisms participating in the aerobic decomposition of organic matter. SOD is also exerted through the aerobic oxidation of reduced species byproducts of anaerobic respiration.

The mass balance, for SOD alone is …


Dollar Bay – Oxygen Calculations

For the SOD calculation, we need to choose the interval of oxygen depletion.

(the question of the cones)


Time-Variable vt

Values for Onondaga Lake (Doerr et al. 1996)


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