Analytical Dissolved Oxygen Model for Sinusoidally Varying BOD
Publication: Journal of Hydrologic Engineering
Volume 2, Issue 4
Abstract
An analytical solution is developed for biochemical oxygen demand (BOD) and dissolved oxygen concentration (DO) in a river for the case when BOD loading concentration and DO loading concentration each vary sinusoidally. The analytical solution is for the quasi-steady-state (periodic) case as the initial condition is ignored, but the boundary conditions are satisfied. Thus, the analytical solution is applicable from the BOD and DO loading locations to the concentration front. This distance is calculated by making use of the memory length concept. The solution is developed by breaking the BOD and DO equations into three subproblems, which are solved separately, then combined by applying the superposition principle. The solution is tested against published solutions from the literature, which include an integral equation, a perturbation solution, and a numerical solution. The analytical solution is easy to apply and extends the cases with which engineers can compare their numerical models. The results suggest the importance of considering time variable wastewater loadings to a river when designing water quality monitoring networks.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Adrian, D. D., Yu, F. X., and Barbe, D.(1994). “Water quality modeling for a sinusoidally varying waste discharge concentration.”Water Res., 28(5), 1167–1174.
2.
Alshawabkeh, A. N., and Adrian, D. D.(1997). “Analytical water quality model for a sinusoidally varying BOD discharge concentration.”Water Res., 31(5), 1207–1215.
3.
Bella, D. A., and Dobbins, W. E.(1968). “Difference modeling of stream pollution.”J. Sanit. Engrg. Div., ASCE, 94(5), 995–1016.
4.
Bennett, J. P.(1971). “Convolution approach to the solution for the dissolved oxygen balance equation in a stream.”Water Resour. Res., 7(3), 580–590.
5.
Carslaw, H. S., and Jaeger, J. C. (1959). Conduction of heat in solids. Oxford University Press, New York, N.Y.
6.
Carslaw, H. S., and Jaeger, J. C. (1963). Operational methods in applied mathematics. Dover Publications, New York, N.Y.
7.
Dresnack, R., and Dobbins, W. E.(1968). “Numerical analysis of BOD and DO profiles.”J. Sanit. Engrg. Div., ASCE, 94(5), 789–807.
8.
Fair, G. M.(1939). “The dissolved oxygen sag—an analysis.”Sewage Works J., 11, 445.
9.
Franklin, P. (1958). Functions of complex variables. Prentice-Hall, Englewood Cliffs, N.J.
10.
Harleman, D. R. F., Holley, E. R., and Huber, W. C. (1966). “Interpretation of water polution data from tidal estuary models.”Proc., 3rd Int. Conf. on Water Pollution Res., Pergamon Press, Inc., Tarrytown, N.Y.
11.
Holley, E. A.(1969). “Discussion of `Difference modeling of stream pollution,' by D. A. Bella and W. E. Dobbins.”J. Sanit. Engrg. Div., ASCE, 95(5), 968–972.
12.
Holley, E. R., and Harleman, D. R. F. (1965). “Dispersion of pollutants in estuary-type flows.”Hydrodyn. Lab. Rep. 74, Massachusetts Inst. of Technol., Cambridge, Mass.
13.
Kirkham, D., and Powers, W. L. (1972). Advanced soil physics. Wiley-Interscience, New York, N.Y.
14.
Koussis, A. D., Kokitar, P., and Menta, R.(1990). “Modeling DO concentrations in streams with dispersion.”J. Envir. Engrg. Div., ASCE, 116(3), 601–614.
15.
Li, W. H.(1972). “Effects of dispersion on DO-SAG in uniform flow.”J. Sanit. Engrg. Div., ASCE, 98(1), 169–182.
16.
Liukov, R. V. (1968). Analytical heat diffusion theory, J. P. Hartnett, ed., Academic Press, New York, N.Y.
17.
McBride, G. B., and Rutherford, J. C.(1984). “Accurate modeling of river pollutant transport.”J. Envir. Engrg. Div., ASCE, 110(4), 809–827.
18.
Resources Agency of California, Dept. of Water Resour. (1962). Sacramento Water Pollution Survey, Bull. no. 111, and Appendix B.
19.
Sanders, T. G., Ward, R. C., Loftis, J. C., Steele, T. D., Adrian, D. D., and Yevjevich, V. (1983). Design of networks for monitoring water quality. Water Resources Publications, Littleton, Colo.
20.
Siemons, J. (1970). “Numerical methods for the solution of diffusion-advection equations.”Rep. 88, Delft Hydr. Lab., Delft, The Netherlands.
21.
Streeter, W. H., and Phelps, E. B. (1925). “A study of the pollution and natural purification of the Ohio River, III—factors concerned in the phenomena of oxidation and reaeration.”Public Health Bull. 146, U.S. Public Health Service, Government Printing Office, Washington, D.C.
22.
Theriault, E. J. (1927). “The dissolved oxygen demand of polluted waters.”Public Health Bull. 173, U.S. Public Health Service, Government Printing Office, Washington, D.C.
23.
Thomann, R. V. (1974). Systems analysis and water quality management. McGraw-Hill Book Co., Inc., New York, N.Y.
24.
Thomann, R. V., and Mueller, J. A. (1987). Principles of surface water quality modeling and control. Harper & Row, Publishers, Inc., New York, N.Y.
25.
Thomas, H. A.(1948). “Pollution load capacity of streams.”Water and Sewage Works, 95(11), 409.
26.
Tikhonov, A. N., and Samarskii, A. A. (1963). Equations of mathematical physics, D. M. Brink, ed., Pergamon Press, New York, N.Y.
27.
van Genuchten, M. T., and Alves, W. J. (1982). “Analytical solutions of the one-dimensional convective-dispersive solute transport equation.”Tech. Bull. 1661, U.S. Dept. of Agr., Agric. Res. Service.
28.
Wang, C., Sun, N., and Yeh, W. W.(1986). “An upstream weight multiple-cell balance finite-element method for solving three-dimensional convection-dispersion equations.”Water Resour. Res., 22(11), 1575–1589.
29.
Yeh, G. T.(1986). “An orthogonal-upstream finite element approach to modeling aquifer contaminant transport.”Water Resour. Res., 22(6), 952–964.
30.
Yu, F. X., Adrian, D. D., and Singh, V. P.(1991). “Modeling river quality by the superposition method.”J. Envir. Sys., 20(4), 1–16.
Information & Authors
Information
Published In
Journal of Hydrologic Engineering
Volume 2 • Issue 4 • October 1997
Pages: 180 - 187
Copyright
Copyright © 1997 American Society of Civil Engineers.
History
Published online: Oct 1, 1997
Published in print: Oct 1997
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.