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
∂ D ∂ t = k 1 L t − k 2 D {\displaystyle {\frac {\partial D}{\partial t}}=k_{1}L_{t}-k_{2}D}
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
D = k 1 L a k 2 − k 1 ( e − k 1 t − e − k 2 t ) + D a e − k 2 t {\displaystyle D={\frac {k_{1}L_{a}}{k_{2}-k_{1}}}(e^{-k_{1}t}-e^{-k_{2}t})+D_{a}e^{-k_{2}t}}
where
D {\displaystyle D} is the saturation deficit, which can be derived from the dissolved oxygen concentration at saturation minus the actual dissolved oxygen concentration ( D = D O s a t − D O {\displaystyle D=DO_{sat}-DO} ). D {\displaystyle D} has the dimensions g m 3 {\displaystyle {\tfrac {\mathrm {g} }{\mathrm {m} ^{3}}}} .
k 1 {\displaystyle k_{1}} is the deoxygenation rate, usually in d − 1 {\displaystyle d^{-1}} .
k 2 {\displaystyle k_{2}} is the reaeration rate, usually in d − 1 {\displaystyle d^{-1}} .
L a {\displaystyle L_{a}} is the initial oxygen demand of organic matter in the water, also called the ultimate BOD (BOD at time t=infinity). The unit of L a {\displaystyle L_{a}} is g m 3 {\displaystyle {\tfrac {\mathrm {g} }{\mathrm {m} ^{3}}}} .
L t {\displaystyle L_{t}} is the oxygen demand remaining at time t, L t = L a e − k 1 t {\displaystyle L_{t}=L_{a}e^{-k_{1}t}} .
D a {\displaystyle D_{a}} is the initial oxygen deficit [ g m 3 ] {\displaystyle [{\tfrac {\mathrm {g} }{\mathrm {m} ^{3}}}]} .
t {\displaystyle t} is the elapsed time, usually [ d ] {\displaystyle [d]} .
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