Comparative Review of Longitudinal Dispersion Coefficient Equations in Rivers
Publication: Journal of Environmental Engineering
Volume 147, Issue 9
Abstract
This paper presents analysis and estimation of the longitudinal dispersion coefficient, a key hydrologic parameter for transport of contaminants in rivers and streams. The longitudinal dispersion coefficient varies spatially in streams with changes in the hydrologic parameters, e.g., the cross-sectional width, depth, sinuosity, and velocity of flow. Many theoretical and empirical equations are reported in the literature. After a comprehensive review, 30 equations for prediction of the longitudinal dispersion coefficient were selected from published research. These equations were used in this analysis. Hydrologic data from 59 river reaches were used for estimation of the dispersion coefficient. The estimated values of the dispersion coefficient were compared statistically and graphically with the observed values. Results showed that sinuosity significantly impacts estimation of the dispersion coefficient. Computations that include sinuosity improve the performance of the dispersion equation. Observed and estimated values were compared, and the equations were ranked based on the accuracy of estimation. The nine top-ranked equations were used for field validation using the ICWater model. The Sahay equation provides the best results when used in the calculation of concentration in the advection dispersion equation.
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Data Availability Statement
All data generated or used during the study appear in the published article.
Acknowledgments
This study was funded by the Defense Threat Reduction Agency as part of the development of a waterborne transport model for the Department of Defense.
References
Ahsan, N. 2004. “An integrated approach for modeling of water quality of an urban river reach.” Ph.D. thesis, Dept. of Civil Engineering, Indian Institute of Technology Delhi.
Ahsan, N. 2007. “A comparative study of models for longitudinal dispersion coefficient in natural streams.” In Proc., Int. Conf. on Environmental Science and Technology. Houston: American Academy of Sciences.
Bansal, M. K. 1971. “Dispersion in natural streams.” J. Hydraul. Div. 97 (11): 1867–1886. https://doi.org/10.1061/JYCEAJ.0003142.
Bashitialshaaer, R., L. Bengtsson, M. Larson, K. M. Persson, M. Aljaradin, and A.-I. Hossam. 2011. “Sinuosity effects on longitudinal dispersion coefficient.” Int. J. Sustainable Water Environ. Syst. 2 (2): 77–84.
Boye, B. A. 2014. “Integrated modelling of hydrodynamic processes, faecal indicator organisms and related parameters with improved accuracy using parallel (GPU) computing.” Ph.D. thesis, Dept. of Architectural, Civil and Environmental Engineering, Cardiff Univ.
Chapra, S. C. 1977. Surface water-quality modeling. New York: McGraw-Hill.
Deng, Z.-Q., V. P. Singh, and L. Bengtsson. 2001. “Longitudinal dispersion coefficient in straight rivers.” J. Hydraul. Eng. 127 (11): 919–927. https://doi.org/10.1061/(ASCE)0733-9429(2001)127:11(919).
Elder, J. W. 1959. “The dispersion of marked fluid in turbulent shear flow.” J. Fluid Mech. 5: 544–560. https://doi.org/10.1017/S0022112059000374.
Etemad-Sahidi, A. E., and M. Taghipour. 2012. “Predicting longitudinal dispersion coefficient in natural streams using M5′ model tree.” J. Hydraul. Eng. 138 (6): 542–554. https://doi.org/10.1061/(ASCE)HY.1943-7900.0000550.
Falconer, R. A., and M. Hartnett. 1993. “Mathematical modelling of flow, pesticide and nutrient transport for fish-farm planning and management.” Ocean Coastal Manage. 19 (1): 37–57. https://doi.org/10.1016/0964-5691(93)90075-A.
Fischer, H. B. 1967a. “Dispersion predictions in natural streams.” J. Sanitary Eng. Div. 94 (5): 927–943.
Fischer, H. B. 1967b. “The mechanics of dispersion in natural streams.” J. Hydraul. Div. 93 (6): 187–216. https://doi.org/10.1061/JYCEAJ.0001706.
Fischer, H. B. 1973. “Longitudinal dispersion and turbulent mixing in open-channel flow.” Annu. Rev. Fluid Mech. 5 (1): 59–78. https://doi.org/10.1146/annurev.fl.05.010173.000423.
Fischer, H. B. 1975. “Discussion of ‘Simple method for predicting dispersion in streams’.” J. Environ. Eng. Div. 101 (3): 453–455. https://doi.org/10.1061/JEEGAV.0000360.
Fischer, H. B., E. J. List, R. C. Y. Koh, J. Imberger, and N. H. Brooks. 1979. Mixing in inland and coastal waters. New York: Elsevier.
Glover, R. E. 1964. Dispersion of dissolved or suspended materials in flowing streams. Washington, DC: US Government Printing Office.
Godfrey, R. G., and B. J. Frederick. 1970. Stream dispersion at selected sites. Washington, DC: USGS.
Guymer, I. 2002. A national database of travel time, dispersion and methodologies for the protection of river abstractions. Bristol, UK: Environment Agency.
Iwasa, Y., and S. Aya. 1991. “Predicting longitudinal dispersion coefficient in open-channel flow.” In Proc., Int. Symp. on Environmental Hydraulics, edited by J. H. W. Lee and Y. K. Cheung, 505–510. Rotterdam, Netherlands: A.A. Balkema.
Jobson, H. E. 1997. “Predicting travel time and dispersion in rivers and streams.” J. Hydraul. Eng. 123 (11): 971–978. https://doi.org/10.1061/(ASCE)0733-9429(1997)123:11(971).
Jobson, H. E. 1996. Prediction of travel time and longitudinal dispersion in rivers and streams. Washington, DC: USGS.
Jobson, H. E. 1999. Prediction of traveltime and longitudinal dispersion in rivers and streams. Washington, DC: USGS.
Kashefipour, S. M., and R. A. Falconer. 2002. “Longitudinal dispersion coefficients in natural streams.” Water Res. 36 (6): 1596–1608. https://doi.org/10.1016/S0043-1354(01)00351-7.
Koussis, A. D., and J. Rodriguez-Mirasol. 1998. “Hydraulic estimation of dispersion coefficient for streams.” J. Hydraul. Eng. 124 (3): 317–320. https://doi.org/10.1061/(ASCE)0733-9429(1998)124:3(317).
Krenkel, P. A. 1960. “Turbulent diffusion and kinetics of oxygen absorption.” Ph.D. dissertation, Dept. of Sanitary Engineering, Univ. of California.
Launay, M., J. Le Coz, B. Camenen, C. Walter, H. Angot, G. Dramais, J.-B. Faure, and M. Coquery. 2015. “Calibrating pollutant dispersion in 1-D hydraulic models of river networks.” J. Hydro-environ. Res. 9 (1): 120–132. https://doi.org/10.1016/j.jher.2014.07.005.
Leopold, L. B., and T. Maddock, Jr. 1953. The hydraulic geometry of stream channels and some physiographic implications. Washington, DC: USGS.
Li, X., H. Liu, and M. Yin. 2013. “Differential evolution for prediction of longitudinal dispersion coefficients in natural streams.” Water Resour. Manage. 27 (15): 5245–5260. https://doi.org/10.1007/s11269-013-0465-2.
Li, Z. H., J. Huang, and J. Li. 1998. “Preliminary study on longitudinal dispersion coefficient for the gorges reservoir.” In Proc., of the 2nd Int. Symp. on Environmental Hydraulics, edited by J. H. W. Lee, and A. W. Jayawardena. Rotterdam, Netherlands: A.A. Balkema.
Liu, H. 1977. “Predicting dispersion coefficient of streams.” J. Environ. Eng. Div. 103 (1): 59–69. https://doi.org/10.1061/JEEGAV.0000605.
Magazine, M. K., S. K. Pathak, and P. K. Pande. 1988. “Effect of bed and side roughness on dispersion in open channels.” J. Hydraul. Eng. 114 (7): 766–782. https://doi.org/10.1061/(ASCE)0733-9429(1988)114:7(766).
Maghrebi, M. F., and M. Givehchi. 2007. “Using non-dimensional velocity curves for estimation of longitudinal dispersion coefficient.” In Proc., 7th Int. Symp. River Engineering, 87–96. Ahwaz, Iran.
McQuivey, R. S., and T. N. Keefer. 1974. “Simple method for predicting dispersion in streams.” J. Environ. Eng. Div. 100 (4): 997–1011. https://doi.org/10.1061/JEEGAV.0000247.
Parker, F. L. 1961. “Eddy diffusion in reservoirs and pipelines.” J. Hydraul. Div. 87 (3): 151–171. https://doi.org/10.1061/JYCEAJ.0000601.
Riahi-Madvar, H., and S. A. Ayyoubzadeh. 2007. Developing an expert system for predicting pollutant dispersion in natural streams, expert systems. Edited by P. Vizureanu. London: IntechOpen. https://doi.org/10.5772/7081.
Sahay, R. R. 2013. “Predicting longitudinal dispersion coefficients in sinuous rivers by genetic algorithm.” J. Hydrol. Hydromech. 61 (3): 214–221. https://doi.org/10.2478/johh-2013-0028.
Sahay, R. R., and S. Dutta. 2009. “Prediction of longitudinal dispersion coefficients in natural rivers using genetic algorithm.” Hydrol. Res. 40 (6): 544–552. https://doi.org/10.2166/nh.2009.014.
Samuels, W. B., D. E. Amstutz, R. Bahadur, and J. M. Pickus. 2006a. “RiverSpill: A national application for drinking water protection.” J. Hydraul. Eng. 132 (4): 393–403. https://doi.org/10.1061/(ASCE)0733-9429(2006)132:4(393).
Samuels, W. B., and R Bahadur, and C. Ziemniak. 2013. “Waterborne transport modeling of radioactivity from the Fukushima nuclear power plant incident.” In Securing water and wastewater systems: Global experiences, edited by R. M. Clark, and S. Hakim. New York: Springer.
Samuels, W. B., R Bahadur, and C. Ziemniak. 2014. “Development and application of the incident command tool for drinking water protection.” Water Environ. J. 29 (1): 1–15.
Samuels, W. B., R. Bahadur, M. Monteith, and D. E. Amstutz. 2006b. “Using NHDPlus for real-time spill modeling.” In Proc., American Water Resources Association GIS Conf. Woodbridge, VA: American Water Resources Association.
Seo, I., and T. Cheong. 1998. “Predicting longitudinal dispersion coefficient in natural streams.” J. Hydraul. Eng. 124 (1): 25–32. https://doi.org/10.1061/(ASCE)0733-9429(1998)124:1(25).
Seo, I. W., and K. O. Baek. 2004. “Estimation of the longitudinal dispersion coefficient using the velocity profile in natural streams.” J. Hydraul. Eng. 130 (3): 227–236. https://doi.org/10.1061/(ASCE)0733-9429(2004)130:3(227).
Smith, R. 1977. “Long-term dispersion of contaminants in small estuaries.” J. Fluid Mech. 82 (1): 129–146. https://doi.org/10.1017/S0022112077000561.
Sokáč, M., Y. Velísková, and C. Gualtieri. 2018. “An approximate method for 1-D simulation of pollution transport in streams with dead zones.” J. Hydrol. Hydromech. 66 (4): 437–447.
Sokáč, M., Y. Velísková, and C. Gualtieri. 2019. “Application of asymmetrical statistical distributions for 1D simulation of solute transport in streams.” Water 11 (10): 2145. https://doi.org/10.3390/w11102145.
Song, Y. 2017. “Estimating longitudinal dispersion coefficients in natural channels.” Master of Science thesis, Dept. of Civil, Construction, and Environmental Engineering, Iowa State Univ.
Sukhodolov, A. N., V. I. Nikora, P. M. Rowinsky, and W. Czemuszenko. 1997. “A case study of longitudinal dispersion in small lowland rivers.” Water Environ. Res. 69 (7): 1246–1253. https://doi.org/10.2175/106143097X126000.
Tavakollizadeh, A., and S. M. Kashefipur. 2007. “Effects of dispersion coefficient on quality modeling of surface waters.” In Proc., 6th Int. Symp. River Engineering, 67–78. Ahwaz, Iran.
Tayfur, G. 2009. “GA-optimized model predicts dispersion coefficient in natural channels.” Hydrol. Res. 40 (1): 65–78. https://doi.org/10.2166/nh.2009.010.
Tayfur, G., and V. P. Singh. 2005. “Predicting longitudinal dispersion coefficient in natural streams by artificial neural network.” J. Hydraul. Eng. 131 (11): 991–1000. https://doi.org/10.1061/(ASCE)0733-9429(2005)131:11(991).
Taylor, G. I. 1954. “The dispersion of matter in turbulent flow through a pipe.” Proc. R. Soc. London, Ser. A 223 (1155): 446–468.
Tenebe, I. T., A. S. Ogbiye, D. O. Omole, and P. C. Emenike. 2016. “Estimation of longitudinal dispersion coefficient: A review.” Cogent Eng. 3 (1): 1216244. https://doi.org/10.1080/23311916.2016.1216244.
Thackston, E. L. 1966. “Longitudinal mixing and reaeration in natural streams.” Ph.D. dissertation, Dept. of Civil and Environmental Engineering, Vanderbilt Univ.
USEPA. 2015a. Emergency response monitoring data from the Gold King Mine incident. Washington, DC: USEPA.
USEPA. 2015b. Flow and chemistry data from the Gold King Mine Adit, Table 5. Washington, DC: USEPA.
USEPA. 2015c. How did the August 2015 release from the Gold King Mine happen? Washington, DC: USEPA.
USEPA. 2015d. Pollution/situation report 2015 Gold King Mine blowout: Polrep/Sitrep Initial Polrep/Sitrep. Washington, DC: USEPA.
Wang, Y., and W. Huai. 2016. “Estimating the longitudinal dispersion coefficient in straight natural rivers.” J. Hydraul. Eng. 142 (11): 04016048. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001196.
West, J. R., and J. S. Mangat. 1986. “The determination and prediction of longitudinal dispersion coefficients in a narrow, shallow estuary.” Estuarine Coastal Shelf Sci. 22 (2): 161–181. https://doi.org/10.1016/0272-7714(86)90111-3.
White, W. R., H. Milli, and A. D. Crabbe. 1973. Sediment transport: An appraisal of methods. Wallingford, UK: Hydraulics Research Station.
Zeng, Y., and W. Huai. 2014. “Estimation of longitudinal dispersion coefficient in rivers.” J. Hydro-environ. Res. 8 (1): 2–8. https://doi.org/10.1016/j.jher.2013.02.005.
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© 2021 American Society of Civil Engineers.
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Received: Jan 6, 2021
Accepted: Apr 19, 2021
Published online: Jul 9, 2021
Published in print: Sep 1, 2021
Discussion open until: Dec 9, 2021
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