StreeterPhelps Dissolved Oxygen(DO) problem
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 Document type:Math Problem
 Level:Undergraduate
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Introduction
The StreeterPhelps Equation refers to an algebraic equation that was formulated through integration of differential equation that determines oxygen sag. The principle on which the StreeterPhelps Equation is that formulated by StreeterPhelps. The integration of the equation does not follow the procedure that was used by StreeterPhelps, but it involves direct solution by applying finite differences. In most applications, the methods currently used and that derived by StreeterPhelps result into the same answer. The principle on which this model works is based on the theory that when waste water is introduced into a stream, the bacteria in the water utilizes oxygen for various biological activities, resulting into a decrease in oxygen and a corresponding replenishment from the atmosphere. The main difficulty in the use of StreeterPhelps Equation is that it is not defined for a significant number of its equal values of the concentration of oxygen and deoxygenating constants, whereas the numerical method does not result into this difficulty. Therefore, the numerical method has been usually used as a method of predicting the outcomes of the StreeterPhelps model and can be applied for all constant of oxygenation and deoxygenation irrespective of their values.
Background to the Report
The introduction of sewage effluent into a stream results into the production of a biochemical oxygen demand (BOD) that undergoes an exponential decay in time and space. This oxygen demand results into oxygen deficit, or shortage of oxygen. When there is a high oxygen deficit, there is a corresponding increase in replenishment of natural oxygen from the atmosphere into the stream. The processes of the use of oxygen and replenishment that take place concurrently results into oxygen sag curve. Oxygen sag curve refers to a curve that sags in the initial period as a result of demand in oxygen and undergoes asymptomatic recovery downstream as a result of increase in oxygen replenishment.
The differential equation used to determine the DOD is:
^{/v)x}_{d}^{(k}e _{u}/v) D_{d}dD/dx = — (k
In the above equation, x is the distance along the stream, determined downstream of the waste water, v is the velocity of the stream.
The differential sag equation for the dissolved oxygen in water has been used to compare the reduction in oxygen concentration after affluent water has been discharged into a stream. The method of solving this equation can be numerical approach and thus, there is no need for integration. In addition, the differential equation has been valid for various deoxygenation constants.
The StreeterPhelps Equation was derived by H.W. Streeter, who worked as a sanitary engineer in the US Health Service Department in 1925. The derivation of the equation was based on a field study conducted on Ohio River. The study on the river was based on information obtained from April 1914 to May 1915 in the Department of Health Service of the United States. The models developed by StreeterPhelps were improved in 1960s where the use of computers enabled understanding of its contribution to the development of oxygen in the river. Other contributions were made to this model by O’Connor and Thomann in 1960 and 1963 respectively. The contributions were derived from photosynthesis, respiration and SOD. The equation is also referred to as the oxygen sag curve.
The StreeterPhelps Equation is expressed as follows:
^{/v)x}_{o}^{(k}) e_{0}— O _{s}] — (S^{/v)x}_{o}^{(k}— e ^{/v)x}_{d}^{(k}[e _{u})] D_{d}— k _{o}/(k_{d}— [k _{s}O = S
In the above equation, O is the amount of dissolved oxygen and can be expressed as milligrams per Liter (mg/L) at a given distance, x, Ss is the amount of dissolved oxygen under saturation, particular saline conditions, temperature and pressure and Oo is the quantity o dissolved oxygen immediately upstream of the waste water.
Thus, the importance of the above equation is its role in understanding the impact of waste water on the amount of oxygen concentration in the stream compared with the amount of oxygen in the stream before the discharge of the affluent water into it.
Purpose of the Study
This study is aimed at illustrating the application of StreeterPhelps Model ti understand oxygen deficit in rivers and streams after the discharge of affluent water containing wastes into it. This is to be achieved by the assumption that the crosssection of the river along which the affluent water has been discharged is completely stirred reactor that slows down the movement downstream. It is also assumed that the substrate of the biodegradable input is valid in the steady stateconditions. In order to understand the changes in oxygen demand and replenishment in various stages if a river, this study develops the oxygen deficit as a function of time using a mixed bath approach where there is no inflow or outflow with the concentration of the Biodegradable Oxygen Demand (BOD) and oxygen that is dissolved in the water. BOD and the reaeration from the atmosphere are included. The relationship between the changes in oxygen concentration as a result of oxidation is developed.
Plot of the StreeterDo Sag Curve
The expected Critical DO Concentration and critical is that the rate at which oxygen is used will be proportional to the rate at which the substrate undergoes oxidation.
The expected distance of the critical location is 157.12 as shown in the calculation in the excel sheet.


Therefore, Do 
157.1203 

In the above equation:
Do is the amount of initial dissolved oxygen in the river
Qr = Flow rate of the stream upstream of the discharge
Lr = The river’s ultimate BOD
Qw = Flow rate of the stream at a point after the discharge
Lw = The BOD of the discharged water
Figure 1. The StreeterDO Sag Curve for the study
Conclusion
This paper illustrates the procedures followed in the calculation of the possible impacts that waste water can have on the dissolved oxygen in the river. This is achieved by use of the StreeterDO Sag Curve. The curve is used to determine the possible critical DO concentration and the critical time when partially waste water flows into a river. It also illustrates the possible distance and the corresponding critical time. This study is of great importance in understanding the impacts that partially waste water can have on river when the water mixes with that flowing along the river. The StreeterPhelps model illustrated above has the limitation in terms of the accuracy of the materials used to perform the experiments and produce the results. This is because, the instruments may not be accurate and there is the likelihood that the results may not provide the accurate results of oxygen concentration of the waste water. The StreeterPhelps Model illustrated in this paper is based on the assumption that there is an even distribution of BOD input at the cross section of the stream or river along which it flows and it does not mix with water. In addition, the classical StreeterPhelps model used considered only one carbonaceous BOD and one reaeration source. Due to these simplifications, there is the possibility of errors in the model. The model excludes BOD removal due to sedimentation. It assumes that the suspended BOD is converted into a dissolved state, that there is a strong affinity for oxygen in the sediments and that there is the possibility of the impact of photosynthesis and respiration on the oxygen balance of the stream. Furthermore, the accuracy of the results of the experiment is determined by other factors such as the amount of salinity of the water or the environmental temperature under which the experiment is conducted. However, the StreeterPhelps Equation is an important tool that has enabled understanding the pollution levels in rivers so that actions can be taken to prevent the discharge of various forms of waste waters or increasing the level of treatment of the waste waters before discharging them into rivers.
References
Streeter, H. W., and E. B. Phelps. 1925. A study of the pollution and natural purification of the Ohio River. III. Factors concerned in the phenomena of oxidation and reaeration. U.S. Public Health Service, Bulletin No. 146.
Tchobanoglous, G., and E. D. Schroeder. 1984. Water quality: Characteristics, modeling, modification. AddisonWesley, Massachussets.