Abstract
A fundamental feature of the classic Streeter-Phelps system of equations is the lack of a feedback between dissolved oxygen concentration and the rate of organic matter oxidation. This may result in physically incorrect solutions, in which dissolved oxygen concentration is negative. Such physical incorrectness is still potentially hazardous for more complex systems of differential equations, resulting from “evolutionary” complication of the classic version. An example of such solutions is given and a criterion (in the form of an inequality) is derived, providing a sufficient condition for the solution of a classic system to be a priori declared physically incorrect. It is proposed to supplement the classic system by an equation relating the concentration of dissolved oxygen and the oxidation rate of organic matter. The new system, called “closed Streeter-Phelps system” will not yield a physically incorrect solution. It can be recommended as an alternative to the classic system, which is used as the core in many complex models of water quality. An additional effect of such change will be a considerable simplification of the calibration of model parameters and the extension of their application range. Two versions of the closed system are suggested, one of which is proposed to be referred to as the system of Streeter-Phelps-Shishkin.
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Brekhovskikh, V.F., Gidrofizicheskie Factory Formirovaniya Kislorodnogo Rezhima Vodoemov (Hydrophysical Factors of Oxygen Regime Formation in Reservoirs), Moscow: Nauka, 1988.
Brekhovskikh, V.F. and Vol’pyan, G.V., Modeling the Oxygen Regime of the Sukhona River with Allowance Made for Anthropogenic Impact, Vodn. Resur., 1991, no. 4, pp. 198–201.
Buber, A.L., Water Quality Model: An Example of Water Quality Modeling in the Oka River, in Komp’yuternoe modelirovanie v upravlenii vodnymi resursami (Computer Modeling in Water Resources Management), Moscow: Fizmatlit, 2002, pp. 310–322.
Vavilin, V.A., Nelineinye modeli biologicheskoi ochistki i protsessov samoochishcheniya v rekakh (Nonlinear Models of Biological Treatment and Self-Purification Processes in Rivers), Moscow: Nauka, 1983.
Gotovtsev, A.V., Closed Streeter-Phelps System of Equations as a Generalization of the Classic System, in Strategicheskie problemy vodopol’zovaniya Rossii (Strategic Problems in Water Use in Russia), Moscow, 2008, pp. 166–177.
Gotovtsev, A.V., On the Issue of the Application Limits of the System of Differential Equations Implemented in the WQ-Module in MIKE-11 Complex, in Mater. XXXVI shk.-sem. “Matematicheskoe modelirovanie v problemakh ratsional’nogo prirodopol’zovaniya” (Mater. XXXVI Workshop “Mathematical Modeling in the Problems of Rational Nature Development”), Rostovon-Don, 2008, pp. 53–54.
Druzhinin, N.I. and Shishkin, A.I., Matematicheskoe modelirovanie i prognozirovanie zagryazneniya poverkhnostnykh vod sushi, (Mathematical Modeling and Pollution Prediction of Surface Continental Waters), Leningrad: Gidrometeoizdat, 1989.
Kondratyuk, V.G., Oxygen Balance and Reaeration of Streams, Vodn. Resur., 1977, no. 2, pp. 27–40.
Obosnovanie strategii upravleniya vodnymi resursami (Substantiation of Water Resources Management Strategy), Danilova-Danil’yan, V.I., Ed., Moscow: Nauch. mir, 2006.
Pryazhinskaya, V.G., Yaroshevskii, D.M., and Levit-Gurevich, L.K., Komp’yuternoe modelirovanie v upravlenii vodnymi resursami (Computer Modeling in Water Resources Management), Moscow: Fizmatlit, 2002.
Rodziller, I.D., Prognoz kachestva vody vodoemov priemnikov stochnykh vod (Prediction of Water Quality in Water Bodies Receiving Wastewaters), Moscow: Stroiizdat, 1984.
Khranovich, I.L., Upravlenie vodnymi resursami. Potokovye modeli (Water Resources Management: Flow Models), Moscow: Nauch. mir, 2001.
Shishkin, A.I., Osnovy matematicheskogo modelirovaniya konvektivno-diffuzionnogo perenosa primesei (Principles of Mathematical Modeling of Advection-Diffusion Transport of Solutes), Leningrad: LTI TsBP, 1976.
Dobbins, W.E., BOD and Oxygen Relationship in Streams, J. San. Eng. Div. Proc. ASCE, 1964, vol. 90, no. SA3, pp. 53–78.
Loucks, D. and Eelco, Van, B., Water Resources Planning and Management. An Introduction to Methods, Models and Applications, Studies and Reports in Hydrolology, Paris: UNESCO Publ., 2005.
MIKE BASIN. User’s Guide. DHI Water & Environment Software. Copenhagen, 2005.
O’Connor, D.J., Oxygen Balance of an Estuary, J. San. Eng. Div. Proc. ASCE, 1960, vol. 86, no. SA3, pp. 35–55.
Owens, M., Edwards, R.W., and Gibbs, J.W., Some Reaeration Studies in Streams, Intern. J. Air Water Poll., 1964, vol. 8, pp. 469–486.
Streeter, H.W. and Phelps, E.B., A Study of the Pollution and Natural Purification of the Ohio River, U.S. Publ. Health Service Bull., 1925, no. 146, pp. 1–75.
Thomann, R.V., Systems Analysis and Water Quality Management, New York: Environ. Res. Applications Inc, 1972.
Thomas, H. A., Pollution Load Capacity of Streams, Water Sewage Works, 1948, vol. 95, no. 3, pp. 409–413.
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Original Russian Text © A.V. Gotovtsev, 2010, published in Vodnye Resursy, 2010, Vol. 37, No. 2, pp. 250–256.
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Gotovtsev, A.V. Modification of the Streeter-Phelps system with the aim to account for the feedback between dissolved oxygen concentration and organic matter oxidation rate. Water Resour 37, 245–251 (2010). https://doi.org/10.1134/S0097807810020120
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DOI: https://doi.org/10.1134/S0097807810020120