Streeter–Phelps equation
The Streeter–Phelps equation is used in the study of water pollution as a water quality modelling tool. The model describes how dissolved oxygen (DO) decreases in a river or stream along a certain distance by degradation of biochemical oxygen demand (BOD). The equation was derived by H. W. Streeter, a sanitary engineer, and Earle B. Phelps, a consultant for the U.S. Public Health Service, in 1925, based on field data from the Ohio River. The equation is also known as the DO sag equation. Streeter–Phelps equation. The Streeter–Phelps equation determines the relation between the  dissolved oxygen concentration and the biological oxygen demand over time and is a solution to the linear first order differential equation This differential equation states that the total change in oxygen deficit (D) is equal to the difference between the two rates of deoxygenation and reaeration at any time. The Streeter–Phelps equation, assuming a plug-flow stream at steady state is then where formula_7 lies typically within the range 0.05-0.5 formula_8 and formula_9 lies typically within the range 0.4-1.5 formula_8. The Streeter–Phelps equation is also known as the DO sag equation. This is due to the  shape of the graph of the DO over time. Critical oxygen deficit. On the DO sag curve a minimum concentration occurs at some point, along a stream. If the Streeter–Phelps equation is differentiated with respect to time, and set equal to zero, the time at which the minimum DO occurs is expressed by To find the value of the critical oxygen deficit, formula_25, the Streeter–Phelps equation is combined with the equation above, for the critical time, formula_26. Then the minimum dissolved oxygen concentration is Mathematically it is possible to get a negative value of formula_28, even though it  is not possible to have a negative amount of DO in reality. The distance traveled in a river from a given point source pollution or waste discharge downstream to the formula_28 (which is the minimum DO) is found by where formula_31 is the flow velocity of the stream. This formula is a good approximation as long as the flow can be regarded as a plug flow (turbulent). Estimation of reaeration rate. Several estimations of the reaeration rate exist, which generally follow the equation where The constants depend on the system to which the equation is applied, i.e. the flow velocity  and the size of the stream or river. Different values are available in the literature. The software "International Hydrological Programme" applies the following equation derived on the basis of values used in published literature where Temperature correction. Both the deoxygenation rate, formula_7 and reaeration rate, formula_9 can be temperature corrected, following the general formula. where Normally θ has the value 1.048 for formula_7 and 1.024 for formula_9. An increasing temperature has the most impact on the deoxygenation rate, and results in an increased critical deficit (formula_25), and formula_51 decreases. Furthermore, a decreased formula_52 concentration occurs with increasing temperature, which leads  to a decrease in the DO concentration. Mixing of rivers. When two streams or rivers merge or water is discharged to a stream it is possible to determine the BOD and DO after mixing assuming steady state conditions and instantaneous mixing. The two streams are considered as dilutions of each other thus the initial BOD and DO will be and where Numerical approach. Nowadays it is possible to solve the classical Streeter–Phelps equation numerically by use of computers. The differential equations are solved by integration. History. In 1925, a study on the phenomena of oxidation and reaeration in the  Ohio River in the US was published by the sanitary engineer, Harold Warner Streeter and the consultant, Earle Bernard Phelps (1876–1953). The study was based on data obtained from May 1914 to April 1915 by the United States Public Health Service under supervision of Surg. W.H. Frost. More complex versions of the Streeter–Phelps model were introduced during the 1960s, where computers made it possible to include further contributions to the oxygen development in streams. At the head of this development were O'Connor (1960) and Thomann (1963). O'Connor added the contributions from photosynthesis, respiration and sediment oxygen demand (SOD). Thomann  expanded the Streeter–Phelps model to allow for multi segment systems. Applications and limitations. The simple Streeter–Phelps model is based on the assumptions that a single BOD input is distributed evenly at the cross section of a stream or river and that it moves as plug flow with no mixing in the river. Furthermore, only one DO sink (carbonaceous BOD) and one DO source (reaeration) is considered in the classical Streeter–Phelps model. These simplifications will give rise to errors in the model. For example the model does not include BOD removal by sedimentation, that suspended BOD is converted  to a dissolved state, that sediment has an oxygen demand and that photosynthesis and respiration will impact the oxygen balance. Expanded model. In addition to the oxidation of organic matter and the reaeration process, there are many other processes in a stream which affect the DO. In order to make a more accurate model it is possible to include these factors using an expanded model. The expanded model is a modification of the traditional model and includes internal sources (reaeration and photosynthesis) and sinks (BOD, background BOD, SOD and respiration) of DO. It is not always necessary to include all  of these parameters. Instead relevant sources and sinks can be summed to yield the overall solution for the particular model. Parameters in the expanded model can be either measured in the field or estimated theoretically. Background BOD. Background BOD or benthic oxygen demand is the diffuse source of BOD represented by the decay of organic matter that has already settled on the bottom. This will give rise to a constant diffuse input thus the change in BOD over time will be where Sedimentation of BOD. Sedimented BOD does not directly consume oxygen and this should therefore be taken into account . This is done by introducing a rate of BOD removal combined with a rate of oxygen consumption by BOD. Giving a total rate for oxygen removal by BOD where The change in BOD over time is described as where formula_75 is the BOD from organic matter in the water formula_17. formula_87 is typically in the range of 0.5-5 formula_8. Sediment oxygen demand. Oxygen can be consumed by organisms in the sediment. This process is referred to as sediment oxygen demand (SOD). Measurement of SOD can be undertaken by measuring the change of oxygen in a box on the sediment  (benthic respirometer). The change in oxygen deficit due to consumption by sediment is described as where The range of the SOD is typically in the range of 0.1 – 1 formula_96 for a natural river with low pollution and 5 – 10 formula_96 for a river with moderate to heavy pollution. Nitrification. Ammonium is oxidized to nitrate under aerobic conditions Ammonium oxidation can be treated as part of BOD, so that BOD = CBOD + NBOD, where CBOD is the carbonaceous biochemical oxygen demand and NBOD is nitrogenous BOD. Usually CBOD is much higher than the ammonium concentration and thus NBOD often  does not need to be considered. The change in oxygen deficit due to oxidation of ammonium is described as where The range of formula_99 is typically 0.05-0.5 formula_8. Photosynthesis and respiration. Photosynthesis and respiration are performed by algae and by macrophytes. Respiration is also performed by bacteria and animals. Assuming steady state (net daily average) the change in deficit will be where Note that BOD only includes respiration of microorganisms e.g. algae and bacteria and not by macrophytes and animals. Due to the variation of light over time, the variation of the photosynthetic oxygen can be described by a  periodical function over time, where time is after sunrise and before sunset where The range of the daily average value of primary production formula_118 is typically 0.5-10 formula_119.