Formulations of Longitudinal Dispersion Coefficient A Review:

Outline 1.Introduction and Applications 2.Historical background 3.New Formulations 4.Conclusions

Longitudinal ADVECTION + Vertical or Lateral DIFFUSION = Dispersion Coefficient (K)

River Mixing Processes

Practical Applications Pollution Forecasting Pollution Forecasting Temperature Variations in a River Temperature Variations in a River River Water Quality Control River Water Quality Control Re-aeration in Water Bodies Re-aeration in Water Bodies Salt Intrusion in Tidal Estuaries Salt Intrusion in Tidal Estuaries

Laminar flow in a tube of radius a Turbulent flow in a pipe of radius a Historical Background 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) 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 Formulae Discrepancy Ratio Accuracy

Accuracy Comparison Researchers Accuracy (%) Elder (1959) 0.0 Fisher (1975) 37.3 Liu (1977) 67.8 Iwasa & Aya (1991) 54.5

Seo & Cheong (1998) New Formulation Dimensional Analysis: Dimensional Analysis: Factors influencing dispersion Factors influencing dispersion GroupsFactors 1 Fluid Properties Fluid density, Fluid viscosity 2 Hydraulic Characteristics Mean Velocity, Shear Velocity, Width, Depth 3 Geometric Configurations Bed forms, Sinuosity

Seo & Cheong (1998) Buckingham PI Theorem

Neglect irregularities Neglect irregularities Flow is fully turbulent Flow is fully turbulent Simplifications Reduced equation:

Regression Analysis Nonlinear Multi-Regression Equation Linear Multiple Form

Final Equation: Experimental data: 59 streams, 26 states in US 59 streams, 26 states in US 35 used to model equation 35 used to model equation 24 used for verification 24 used for verification

Accuracy: 79% Limitation: Seo & Cheong (1998) Researchers Accuracy (%) Elder (1959) 0.0 Fisher (1975) 37.3 Liu (1977) 67.8 Iwasa & Aya (1991) 54.5

Deng et al. (2002) Lateral Dispersion Coefficient Lateral Dispersion Coefficient Velocity Deviation Parameter Velocity Deviation Parameter Channel Shape Equation Channel Shape Equation Local Flow Depth Local Flow Depth Channel Sinuosity Channel Sinuosity

Deng et al. (2002)

Conclusions Analytical solution by Taylor (1954) available for regular cross sections. Analytical solution by Taylor (1954) available for regular cross sections. Natural streams need empirical means with field data. Natural streams need empirical means with field data. Liu’s equation (1977) offers the best prediction followed by Iwasa & Aya (1991). Liu’s equation (1977) offers the best prediction followed by Iwasa & Aya (1991).

Conclusions Seo and Cheong’s new equation (1998) with field data from 59 streams across 26 states in the US. Seo and Cheong’s new equation (1998) with field data from 59 streams across 26 states in the US. Deng et al. (2002) incorporated the effect of vertical and transverse irregularities. Deng et al. (2002) incorporated the effect of vertical and transverse irregularities.

References 1.Seo, I. W., and Cheong, T. S. (1998). “Predicting Longitudinal Dispersion Coefficient in Natural Streams.” J. Hydr. Engrg., ASCE, 124(1), Liu, H. (1977). “Predicting dispersion coefficient of streams.” J. Envir. Engrg. Div., ASCE, 103(1), Deng, Z.-Q., Bengtsson, L., Singh, V. P., and Adrian, D. D (2002). “Longitudinal Dispersion Coefficient in Single-Channel Streams”, J. Hydr. Engrg., ASCE, 128(10),

Q & A ?