# Presentation: 019019

## Dispersion and Exchanges

## Slide 2

## Diffusion and Dispersion

- Diffusion: process where a constituent moves from a higher concentration to a lower concentration
- Dispersion: mixing caused by physical processes

- time

## Diffusion and Dispersion

- Molecular Diffusion: Random motion of particles
- Eddy Diffusion: Turbulent mixing of particles
- Mechanical Dispersion: mixing caused by variations in velocities

## Mathematical Representations

- Diffusion:
- Fick’s First Law of Diffusion
- Flux =
- Dispersion:
- Analogous to Fick’s First Law

*C**1*

*C**2*

*L**12*

-  Exchange =

## Range of Values for Diffusion and Dispersion

| Process | Direction | Typical Range [m2/s] |
| --- | --- | --- |
| 
Molecular Diffusion | Vertical | 10-8 to 10-9 |
|  | Lateral | 10-8 to 10-9 |
|  | Longitudinal | 10-8 to 10-9 |
| 
Turbulent Diffusion | Vertical | 10-6 to 10-2 |
|  | Lateral | 10-2 to 102 |
|  | Longitudinal | 10-2 to 102 |
| 
Dispersion | Vertical | 10-3 to 10-1 |
|  | Lateral | 10-2 to 100 |
|  | Longitudinal | 10-1 to 104 |

## Diffusion Coefficients

| Condition | Dispersion 
Coefficient [m2/s] |
| --- | --- |
| Molecular Diffusion | 10-9 |
| Compacted Sediment | 10-11 to 10-9 |
| Bioturbated Sediment | 10-9 to 10-8 |
| Lakes – Vertically | 10-6 to 10-4 |
| Large Rivers – Lateral | 10-4 to 10-3 |
| Large Rivers – Longitudinal | 100  to 102 |
| Estuaries – Longitudinal | 102 to 103 |

- From Schnoor, Environmental Modeling: Fate and Transport of Pollutants in Water, Air, and Soil, 1996.

## Determining Dispersion

- Streams & Rivers
- Generally Neglect Dispersion
- Determine by Calibration or Dye Study
- Estuaries
- Calibration to Salinity data using observed downstream boundary concentration as the forcing function
- Lakes
- Calibration to Temperature Data
- Calibration to Chloride Data

## Dispersion in Rivers

**Longitudinal** Dispersion Coefficient in Rivers, *E**x*  [m2/sec]

- Fischer, et al., 1979.

## Dispersion in Rivers

**Lateral** Dispersion Coefficient in Rivers, *E**y*  [m2/sec]

*d* 	= Depth (hydraulic radius), [m]
*u** 	= Shear Velocity, [m/sec]
**	= 0.23 (long, wide lab flume)	Elder, 1959
- 	= 0.17 (straight lab flume)
- 		Sayre (1973), Sayre and Chang (1968)
- 	= 0.22 – 0.65, most 0.3
- 		Yotsukura and Cobb(1972), Yotsukura and Sayre (1976)

- Elder, 1959.

*E**y**  = ****du**

## Dispersion in Rivers

**Vertical** Dispersion Coefficient in Rivers, *E**z*  [m2/sec]

*K**	*= von Karman coefficient, ~ 0.4
*d* 	= Depth (hydraulic radius), [m]
*u** 	= Shear Velocity, [m/sec]

- Jobson and Sayre, 1970.

## Dispersion in Lakes

## Dispersion Measurements

| Chattahoochee River, GA | Ex = 32.5 m2/s |
| --- | --- |
| Susquehanna River, PA | Ex = 92.9 m2/s |
| Missouri River, NB-IA | Ex = 465 – 1487 m2/s |
| Antietam Creek, MD | Ex = 9.3 - 25.6 m2/s |
| Powell River, TN | Ex = 9.5 m2/s |
| Lake Onondaga, NY | Ez = 0.005 – 0.09 cm2/s (across thermocline) |
| Lake Zurich, Switzerland | Ez = 0.02 – 0.71 cm2/s (across thermocline) |
| Lake Erie | Ez = 0.21 cm2/s (across thermocline) |
| Lake Ontario | Ez = 0.063 – 0.0125 cm2/s (across thermocline) |
| Lake Erie | Ez = 15 cm2/s (whole lake) |
| Lake Huron | Ez = 1.16 cm2/s (whole lake) |
| Cayuga Lake | Ez = 2.31 cm2/s (whole lake) |
| Lake Ontario | Ez = 3.47 cm2/s (whole lake) |

## Implementing Dispersion in WASP:Exchanges

*C**1*

*C**2*

*E**12*

## Setting Up Dispersion in WASP

- 1

- 3

- 2

- 4

- 5

- 6

-  Six identical segments. Each 10m x 10m x 10m in size
-  Purely Dispersive System
-  Need to set up exchange between each segment:
-  *E**x* = 10-4 m2/sec

*C**1* = 0.6 mg/L

*C**2* = *C**3* = *C**4* = *C**5* = *C**6* = 0.0 mg/L

## Segment Structure Set Up

- Use Simple Toxicant Module
- Run from
- 1/1/2000 00:00 to 7/1/2000 00:00
- Time Step of 0.1 days
- Print Interval of 1 day
- 6 segments of 10m x 10m x 10m
- Model Conservative Toxicant
- Bypass Solids
- Initial Concentration of 0.6 mg/L in Segment 1

## Segment Tab

- Set up 6 Water Segments

- 1000 m3 each

- 10 m deep

## Set Up Exchanges

## Set Up Exchanges

- Turn on “Surface Water”
- Define “Exchange Function” as “Lateral Dispersion”
- Define Exchanges from Segment one to Segment two
- Define value as 0.0001 m2/sec

## Set Up Exchanges

- WASP Structure of Exchanges requires
-  *E**12* = Dispersion Exchange Coefficient [m2/s]
-  *A**12* = Cross-section Dispersion Area [m2]
-  *L**12* = Characteristic Dispersion Length [m]
-  typically defined as 0.5*(*L**1** + L**2*)
-  where *L**1* and *L**2* are the lengths of
- 	 segments 1 and 2.
- Therefore, Area = 100 m2 and Distance = 10 m

## Set Up Exchanges

## Dispersion Output

## 0	5	15	25	35	45	50 			

- 0	5	15	25	35	45	50

- 600

- 400

- 300

- 200

- 100

- 0

- 500

- Distance

- Concentration [ug/L]