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Dissolved Oxygen Deficit in a Stream

Hardstreeter-phelpsdissolved oxygen
A stream immediately below a discharge has an initial dissolved oxygen deficit of 2.0 mg/L and an ultimate BOD of 20 mg/L. The deoxygenation constant is 0.25 per day and the reaeration constant is 0.45 per day. Using the Streeter-Phelps equation, the deficit one day downstream is most nearly:
  1. A2.4 mg/L
  2. B3.6 mg/L
  3. C4.8 mg/L
  4. D6.2 mg/L
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Answer: C. 4.8 mg/L

The Streeter-Phelps deficit is D = [kd L₀/(kr − kd)](e^(−kd t) − e^(−kr t)) + D₀e^(−kr t). Substituting: [(0.25)(20)/0.20](e^(−0.25) − e^(−0.45)) + 2.0e^(−0.45) = 25(0.7788 − 0.6376) + 1.275 = 3.53 + 1.27 ≈ 4.8 mg/L. Omitting the second term, the decay of the initial deficit, understates the answer by about 1.3 mg/L.
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