Modification of the Streeter-Phelps system with the aim to account for the feedback between dissolved oxygen concentration and organic matter oxidation rate

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.