Formulations ofFormulations of

Longitudinal Dispersion Longitudinal Dispersion CoefficientCoefficient

A A Review:Review:

OutlineOutline

1.1. Introduction and ApplicationsIntroduction and Applications

2.2. Historical backgroundHistorical background

3.3. New FormulationsNew Formulations

4.4. ConclusionsConclusions

Longitudinal Longitudinal ADVECTIONADVECTION

++

Vertical or Lateral Vertical or Lateral DIFFUSIONDIFFUSION

==

Dispersion Coefficient (K)Dispersion Coefficient (K)

River Mixing ProcessesRiver Mixing Processes

Practical ApplicationsPractical Applications

Pollution ForecastingPollution Forecasting

Temperature Variations in a Temperature Variations in a RiverRiver

River Water Quality ControlRiver Water Quality Control

Re-aeration in Water BodiesRe-aeration in Water Bodies

Salt Intrusion in Tidal EstuariesSalt Intrusion in Tidal Estuaries

' '

0 0 0

1 1 -

W y y

t

K hu hu dydydyA h

2 2

*

U W

KhU

1.5

*

2.0K W

hU h

1.5

* 0.18U

U

* 10.1K au

2 20

192

a uK

D

Laminar flow in a tube of radius Laminar flow in a tube of radius aa

Turbulent flow in a pipe of radius Turbulent flow in a pipe of radius aa

Historical BackgroundHistorical Background

* 5.93K hU

Taylor (1953 & 1954)Taylor (1953 & 1954) Elder (1959)Elder (1959)

2 2

*

0.011U W

KhU

Fischer (1966 & 1975)Fischer (1966 & 1975) Liu (1977)Liu (1977) Iwasa and Aya (1991)Iwasa and Aya (1991) Taylor (1953 & 1954)Taylor (1953 & 1954)

Elder (1959)Elder (1959)

Fischer (1966 & 1975)Fischer (1966 & 1975)

Liu (1977)Liu (1977)

Iwasa and Aya (1991)Iwasa and Aya (1991)

Comparison of FormulaeComparison of Formulae

Discrepancy RatioDiscrepancy Ratio

AccuracyAccuracy

log p

m

KDiscrepancy Ratio

K

0.3 0.3Discrepancy Ratio

Accuracy ComparisonAccuracy Comparison

ResearchersResearchers Accuracy (%)Accuracy (%)

Elder (1959)Elder (1959) 0.00.0

Fisher (1975)Fisher (1975) 37.337.3

Liu (1977)Liu (1977) 67.867.8

Iwasa & Aya Iwasa & Aya (1991)(1991) 54.554.5

Seo & Cheong (1998)Seo & Cheong (1998)New FormulationNew Formulation

• Dimensional Analysis:Dimensional Analysis: Factors influencing dispersionFactors influencing dispersion

GroupsGroups FactorsFactors

11 Fluid PropertiesFluid Properties Fluid density, Fluid Fluid density, Fluid viscosityviscosity

22Hydraulic Hydraulic Characteristics Characteristics

Mean Velocity, Shear Mean Velocity, Shear Velocity, Width, DepthVelocity, Width, Depth

33Geometric Geometric ConfigurationsConfigurations Bed forms, SinuosityBed forms, Sinuosity

Seo & Cheong (1998)Seo & Cheong (1998)

Buckingham PI TheoremBuckingham PI Theorem

* *

( , , , , )f n

K Uh U Wf S S

hU U h

• Neglect irregularitiesNeglect irregularities

• Flow is fully turbulentFlow is fully turbulent

SimplificationsSimplifications

& f nS S

Reduced Reduced equation:equation:

Uh

* *

( , )K U W

fhU U h

Regression AnalysisRegression Analysis

Nonlinear Multi-Regression EquationNonlinear Multi-Regression Equation

Linear Multiple FormLinear Multiple Form

1 2 3 pY X X X X

1 2ln ln ln ln

ln lnp

Y X X

X

Final Equation:Final Equation:

Experimental data:Experimental data:• 59 streams, 26 states in US59 streams, 26 states in US• 35 used to model equation35 used to model equation• 24 used for verification24 used for verification

1.428 0.620

* *

5.915K U W

hU U h

Accuracy: 79%Accuracy: 79%

Limitation:Limitation:

Seo & Cheong (1998)Seo & Cheong (1998)

ResearcherResearcherss

Accuracy Accuracy (%)(%)

Elder Elder (1959)(1959) 0.00.0

Fisher Fisher (1975)(1975) 37.337.3

Liu (1977)Liu (1977) 67.867.8

Iwasa & Iwasa & Aya (1991)Aya (1991) 54.554.5

Deng et al. (2002)Deng et al. (2002)

• Lateral Dispersion CoefficientLateral Dispersion Coefficient

• Velocity Deviation ParameterVelocity Deviation Parameter

• Channel Shape EquationChannel Shape Equation

• Local Flow DepthLocal Flow Depth

• Channel SinuosityChannel Sinuosity

Deng et al. (2002)Deng et al. (2002)

1 5/ 20 * * *0 0 0

5/ 2* *0 0

C

C/ x

I h F h h Fd d

Fh h d d d

x

2 2

* 0

* * *

I IK U W

U h M U h

1*

* **max 0

UI h d

u

1.38

* *

10.145

3520

U WM

U h

ConclusionsConclusions

• Analytical solution by Taylor Analytical solution by Taylor (1954) available for regular cross (1954) available for regular cross sections.sections.

• Natural streams need empirical Natural streams need empirical means with field data.means with field data.

• Liu’s equation (1977) offers the Liu’s equation (1977) offers the best prediction followed by Iwasa best prediction followed by Iwasa & Aya (1991).& Aya (1991).

ConclusionsConclusions

• Seo and Cheong’s new equation Seo and Cheong’s new equation (1998) with field data from 59 (1998) with field data from 59 streams across 26 states in the streams across 26 states in the US.US.

• Deng et al. (2002) incorporated Deng et al. (2002) incorporated the effect of vertical and the effect of vertical and transverse irregularities.transverse irregularities.

ReferencesReferences

1.1. Seo, I. W., and Cheong, T. S. (1998). Seo, I. W., and Cheong, T. S. (1998). “Predicting Longitudinal Dispersion “Predicting Longitudinal Dispersion Coefficient in Natural Streams.” Coefficient in Natural Streams.” J. Hydr. J. Hydr. Engrg.Engrg., ASCE, 124(1), 25-32., ASCE, 124(1), 25-32.

2.2. Liu, H. (1977). “Predicting dispersion Liu, H. (1977). “Predicting dispersion coefficient of streams.” coefficient of streams.” J. Envir. Engrg. Div.J. Envir. Engrg. Div., , ASCE, 103(1), 59-69.ASCE, 103(1), 59-69.

3.3. Deng, Z.-Q., Bengtsson, L., Singh, V. P., and Deng, Z.-Q., Bengtsson, L., Singh, V. P., and Adrian, D. D (2002). “Longitudinal Dispersion Adrian, D. D (2002). “Longitudinal Dispersion Coefficient in Single-Channel Streams”, Coefficient in Single-Channel Streams”, J. J. Hydr. Engrg.Hydr. Engrg., ASCE, 128(10), 901-916., ASCE, 128(10), 901-916.

Q & A ?Q & A ?