Dispersion
Dispersion and ExchangesWatershed & Water Quality Modeling
Technical Support Center
Watershed & Water Quality Modeling Technical Support
CenterDiffusion and DispersionDiffusion: process where a
constituent moves from a higher concentration to a lower
concentration
Dispersion: mixing caused by physical processestimeWatershed
& Water Quality Modeling Technical Support CenterDiffusion and
DispersionMolecular Diffusion: Random motion of particles
Eddy Diffusion: Turbulent mixing of particles
Mechanical Dispersion: mixing caused by variations in
velocities
Watershed & Water Quality Modeling Technical Support
CenterMathematical RepresentationsDiffusion: Ficks First Law of
Diffusion Flux = Dispersion:Analogous to Ficks First Law
C1C2L12
Exchange =Watershed & Water Quality Modeling Technical
Support CenterRange of Values for Diffusion and Dispersion
ProcessDirectionTypical Range [m2/s]Molecular
DiffusionVertical10-8 to 10-9Lateral10-8 to 10-9Longitudinal10-8 to
10-9Turbulent DiffusionVertical10-6 to 10-2Lateral10-2 to
102Longitudinal10-2 to 102DispersionVertical10-3 to 10-1Lateral10-2
to 100Longitudinal10-1 to 104Watershed & Water Quality Modeling
Technical Support CenterDiffusion CoefficientsConditionDispersion
Coefficient [m2/s]Molecular Diffusion10-9Compacted Sediment10-11 to
10-9Bioturbated Sediment10-9 to 10-8Lakes Vertically10-6 to
10-4Large Rivers Lateral10-4 to 10-3Large Rivers Longitudinal100 to
102Estuaries Longitudinal102 to 103From Schnoor, Environmental
Modeling: Fate and Transport of Pollutants in Water, Air, and Soil,
1996.Watershed & Water Quality Modeling Technical Support
CenterDetermining DispersionStreams & RiversGenerally Neglect
DispersionDetermine by Calibration or Dye StudyEstuariesCalibration
to Salinity data using observed downstream boundary concentration
as the forcing functionLakesCalibration to Temperature
DataCalibration to Chloride Data
Watershed & Water Quality Modeling Technical Support
CenterDispersion in RiversLongitudinal Dispersion Coefficient in
Rivers, Ex [m2/sec]
= Mean Velocity [m/sec]B= Width, [m]d = Depth (hydraulic
radius), [m]u* = Shear Velocity, [m/sec]= S= Channel Slope
[m/m]
Fischer, et al., 1979. Watershed & Water Quality Modeling
Technical Support CenterDispersion in RiversLateral Dispersion
Coefficient in Rivers, Ey [m2/sec]
d = Depth (hydraulic radius), [m]u* = Shear Velocity, [m/sec]f=
0.23 (long, wide lab flume)Elder, 1959= 0.17 (straight lab flume)
Sayre (1973), Sayre and Chang (1968)= 0.22 0.65, most 0.3Yotsukura
and Cobb(1972), Yotsukura and Sayre (1976)
Elder, 1959. Ey = fdu*Watershed & Water Quality Modeling
Technical Support CenterDispersion in RiversVertical Dispersion
Coefficient in Rivers, Ez [m2/sec]
K= von Karman coefficient, ~ 0.4d = Depth (hydraulic radius),
[m]u* = Shear Velocity, [m/sec]Jobson and Sayre, 1970.
Watershed & Water Quality Modeling Technical Support
CenterDispersion in Lakes
Vertical Dispersion Coefficient in Lakes, Ez [m2/sec] d = Depth,
[m]Jobson and Sayre, 1970.
Watershed & Water Quality Modeling Technical Support
CenterDispersion MeasurementsChattahoochee River, GAEx = 32.5
m2/sSusquehanna River, PAEx = 92.9 m2/sMissouri River, NB-IAEx =
465 1487 m2/sAntietam Creek, MDEx = 9.3 - 25.6 m2/sPowell River,
TNEx = 9.5 m2/sLake Onondaga, NYEz = 0.005 0.09 cm2/s (across
thermocline)Lake Zurich, SwitzerlandEz = 0.02 0.71 cm2/s (across
thermocline)Lake ErieEz = 0.21 cm2/s (across thermocline)Lake
OntarioEz = 0.063 0.0125 cm2/s (across thermocline)Lake ErieEz = 15
cm2/s (whole lake)Lake HuronEz = 1.16 cm2/s (whole lake)Cayuga
LakeEz = 2.31 cm2/s (whole lake)Lake OntarioEz = 3.47 cm2/s (whole
lake)Watershed & Water Quality Modeling Technical Support
CenterImplementing Dispersion in WASP:ExchangesC1C2E12Watershed
& Water Quality Modeling Technical Support CenterSetting Up
Dispersion in WASP132456 Six identical segments. Each 10m x 10m x
10m in size
Purely Dispersive System
Need to set up exchange between each segment: Ex = 10-4 m2/secC1
= 0.6 mg/LC2 = C3 = C4 = C5 = C6 = 0.0 mg/LWatershed & Water
Quality Modeling Technical Support CenterSegment Structure Set
UpUse Simple Toxicant ModuleRun from 1/1/2000 00:00 to 7/1/2000
00:00Time Step of 0.1 daysPrint Interval of 1 day6 segments of 10m
x 10m x 10mModel Conservative ToxicantBypass SolidsInitial
Concentration of 0.6 mg/L in Segment 1Watershed & Water Quality
Modeling Technical Support CenterSegment Tab
Set up 6 Water Segments1000 m3 each10 m deepWatershed &
Water Quality Modeling Technical Support CenterSet Up Exchanges
Watershed & Water Quality Modeling Technical Support
Center
Set Up ExchangesTurn on Surface Water
Define Exchange Function as Lateral Dispersion
Define Exchanges from Segment one to Segment two
Define value as 0.0001 m2/secWatershed & Water Quality
Modeling Technical Support CenterSet Up Exchanges
WASP Structure of Exchanges requires E12 = Dispersion Exchange
Coefficient [m2/s] A12 = Cross-section Dispersion Area [m2] L12 =
Characteristic Dispersion Length [m] typically defined as 0.5*(L1 +
L2) where L1 and L2 are the lengths of segments 1 and 2.
Therefore, Area = 100 m2 and Distance = 10 mWatershed &
Water Quality Modeling Technical Support Center
Set Up ExchangesWatershed & Water Quality Modeling Technical
Support CenterDispersion Output
Watershed & Water Quality Modeling Technical Support
Center
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